Lecture Notes for Logic J. Dmitri Gallow Contents 1 Basic Concepts of Logic 4 1.1 Finding Argumentative Structure ...................................... 5 1.2 Conditionals ................................................. 7 1.2.1 Necessary and Sufficient Conditions ................................ 7 1.3 Deductive Validity .............................................. 8 1.4 Inductive Strength .............................................. 9 2 Basic Concepts of Logic, Day 2 10 2.0.1 Review ................................................ 10 2.1 Proving Invalidity, take 1 ........................................... 10 2.1.1 Venn Diagrams ........................................... 11 2.1.2 Venn Diagrams, Counterexamples, and Validity .......................... 12 3 Basic Concepts of Logic, Day 3 15 3.1 Review .................................................... 15 3.2 Formal Deductive Validity .......................................... 15 3.3 Proving Invalidity, take 2 ........................................... 17 3.4 Formal Deductive Invalidity ......................................... 18 4 Dialectics 21 4.1 Rules for Dialectics .............................................. 23 4.1.1 Example: A Failed Dialectic ..................................... 25 5 Informal Fallacies 27 5.1 Fallacies of Irrelevance ............................................ 28 5.1.1 Argument Against the Person (Ad Hominem) ........................... 28 5.1.2 Straw Man .............................................. 29 5.1.3 Appeal to Force (Ad Baculum) ................................... 29 5.1.4 Appeal to the People (Ad Populum) ................................ 30 5.1.5 Appeal to Ignorance (Ad Ignorantiam) ............................... 30 5.1.6 Red Herring (Ignoratio Elenchi) .................................. 31 6 Informal Fallacies, Day 2 32 6.1 Fallacies Involving Ambiguity ........................................ 32 6.1.1 Equivocation ............................................. 32 6.1.2 Amphiboly .............................................. 33 6.1.3 Composition/Division ........................................ 33 6.2 Fallacies Involving Unwarranted Assumptions ............................... 34 6.2.1 Begging the Question (Petitio Principii) .............................. 34 6.2.2 False Dilemma ............................................ 34 6.2.3 False Cause Fallacy ......................................... 35 1 7 Propositional Logic 37 7.1 Syntax for PL ................................................. 37 7.1.1 Vocabulary .............................................. 37 7.1.2 Grammar .............................................. 38 7.1.3 Main Operators and Subformulae ................................. 39 7.2 Semantics for PL ............................................... 40 7.2.1 The Meaning of the Statement Letters ............................... 41 7.2.2 The Meaning of ‘∼’ ......................................... 41 7.2.3 The Meaning of ‘•’ .......................................... 41 7.2.4 The Meaning of ‘_’ ......................................... 41 7.2.5 The Meaning of ‘⊃’ ......................................... 42 7.2.6 The Meaning of ‘≡’ ......................................... 42 7.2.7 Determining the Truth-value of a wff of PL ............................ 42 7.3 Translation from PL to English ....................................... 44 7.4 Translation from English to PL ....................................... 46 7.4.1 Negation ............................................... 46 7.4.2 Conjunction ............................................. 47 7.4.3 Disjunction ............................................. 47 7.4.4 The Material Conditional and Biconditional ............................ 48 8 Logical Notions of PL: Validity 49 8.1 How to Construct a Truth-Table ....................................... 49 8.2 What a Truth-Table Represents ....................................... 51 8.3 PL-Validity ................................................. 52 9 Propositional Logic Derivations, day 1 56 9.1 The Basics ................................................... 56 9.2 Rules of Implication ............................................. 57 9.2.1 Modus Ponens ............................................ 57 9.2.2 Modus Tollens ............................................ 57 9.2.3 Hypothetical Syllogism ....................................... 58 9.2.4 Disjunctive Syllogism ........................................ 58 9.2.5 Simplification ............................................ 60 9.2.6 Conjunction ............................................. 60 9.2.7 Addition ............................................... 61 9.2.8 Constructive Dilemma ....................................... 61 9.3 A Mistake to Avoid .............................................. 63 9.4 Rules of Replacement ............................................ 63 9.4.1 De Morgan’s ............................................. 64 9.4.2 Commutativity ........................................... 64 9.4.3 Associativity ............................................. 65 9.4.4 Distribution ............................................. 66 9.4.5 Double Negation ........................................... 67 9.4.6 Transposition ............................................ 68 9.4.7 Material Implication ......................................... 69 9.4.8 Material Equivalence ........................................ 69 9.4.9 Exportation ............................................. 70 10 Propositional Logic Derivations, Day 2 72 10.1 Four Final Rules of Inference ........................................ 72 10.1.1 Subderivations ............................................ 72 10.1.2 Conditional Proof .......................................... 75 10.1.3 Indirect Proof ............................................ 76 10.2 PL-Derivability and the Logical Notions of PL .............................. 78 2 10.2.1 Some New Notation ......................................... 78 10.2.2 PL-Validity ............................................. 78 10.2.3 PL-Tautologies and PL-Self-Contradictions ........................... 79 10.2.4 PL-Equivalence and PL-Contradiction .............................. 81 10.2.5 PL-Inconsistency .......................................... 83 11 Categorical Propositions 85 11.1 Categorical Propositions ........................................... 85 11.1.1 The Components of a Categorical Proposition ........................... 86 11.1.2 Quality, Quantity, and Distribution ................................. 87 11.2 The Square of Opposition .......................................... 89 11.3 Categorical Syllogisms ............................................ 93 11.3.1 Major Terms, Minor Terms, Middle Terms ............................. 94 11.3.2 Mood and Figure .......................................... 94 11.4 Venn Diagrams ................................................ 96 11.4.1 Representing Standard Form Categorical Propositions with Venn Diagrams (Modern Under- standing) ............................................... 97 11.4.2 Testing the Validity of Categorical Syllogisms with Venn Diagrams (Modern Understanding) . 98 12 Quantificational Logic: Syntax and Semantics 101 12.1 The Language QL .............................................. 101 12.1.1 The Syntax of QL .......................................... 102 12.1.2 Semantics for QL .......................................... 107 12.2 Translations from QL into English ..................................... 110 12.2.1 Translating Simple Quantified wffs of QL ............................. 110 12.2.2 Translating More Complicated Quantified wffs of QL ....................... 110 12.3 Translations from English into QL ..................................... 112 13 Quantificational Logic: Validity, Equivalence, and the Rest 114 13.1 Notation ................................................... 114 13.2 QL-Validity ................................................. 114 13.3 QL-tautologies, QL-self-contradictions, & QL-contingencies ...................... 116 13.4 QL-Equivalence & QL-Contradiction ................................... 117 13.5 QL-Consistency & QL-Inconsistency ................................... 119 14 Quantificational Logic: Derivations 121 14.1 Substitution Instances ............................................ 121 14.2 QL-Derivations ............................................... 122 14.2.1 New Rules of Replacement ..................................... 122 14.2.2 New Rules of Implication ...................................... 123 14.2.3 Final Thoughts ............................................ 131 14.3 QL-Derivability and the Logical Notions of QL .............................. 132 14.3.1 QL-Validity ............................................. 133 14.3.2 QL-Tautologies and QL-Self-Contradictions ........................... 133 14.3.3 QL-Equivalence and QL-Contradiction .............................. 134 14.3.4 QL-Inconsistency .......................................... 135 15 Quantificational Logic with Identity 137 15.1 The Language QLI .............................................. 137 15.1.1 Syntax for QLI ........................................... 137 15.1.2 Semantics for QLI ......................................... 138 15.2 QLI Derivations ............................................... 139 15.3 Translations from English to QLI ...................................... 143 15.3.1 Number Claims ........................................... 143 15.4 The Only ..................................................