An Introduction to Formal Logic Second edition Peter Smith February 27, 2019
Peter Smith 2018. Not for re-posting or re-circulation. Comments and corrections please to ps218 at cam dot ac dot uk
1 What is deductive logic? 1 1.1 What is an argument? 1 1.2 Kinds of evaluation 1 1.3 Deduction vs. induction 2 1.4 Just a few more examples 4 1.5 Generalizing 5 1.6 Summary 7 Exercises 1 7 2 Validity and soundness 9 2.1 Validity defined 9 2.2 Consistency, validity, and equivalence 11 2.3 Validity, truth, and the invalidity principle 12 2.4 Inferences and arguments 14 2.5 Valid vs true 15 2.6 What s the use of deduction? 15 2.7 An illuminating circle? 17 2.8 Summary 18 Exercises 2 18 3 Forms of inference 20 3.1 More forms of inference 20 3.2 Four basic points about the use of schemas 22 3.3 Arguments can instantiate many patterns 24 3.4 Summary 26 Exercises 3 26 4 Proofs 27 4.1 Proofs: first examples 27 4.2 Fully annotated proofs 28 4.3 Glimpsing an ideal 30 4.4 Deductively cogent multi-step arguments 31 4.5 Indirect arguments 33 4.6 Summary 35 Exercises 4 36
ii 5 The counterexample method 37 5.1 But you might as well argue... 37 5.2 The counterexample method, more carefully 38 5.3 A quantifier shift fallacy 39 5.4 Summary 41 Exercises 5 41 6 Logical validity 42 6.1 Topic neutrality 42 6.2 Logical validity, at last 43 6.3 Logical necessity 45 6.4 The boundaries of logical validity? 45 6.5 Definitions of validity as rational reconstructions 46 6.6 Summary 48 Exercises 6 48 7 Propositions and forms 49 7.1 Types vs tokens 49 7.2 Sense vs tone 49 7.3 Are propositions sentences? 50 7.4 Are propositions truth-relevant contents? 51 7.5 Why we can be indecisive 52 7.6 Forms of inference again 53 7.7 Summary 54 Interlude: From informal to formal logic 55 8 Three connectives 57 8.1 Two simple arguments 57 8.2 And 58 8.3 Or 59 8.4 Not 61 8.5 Scope 62 8.6 Formalization 63 8.7 The design brief for PL languages 64 8.8 One PL language 66 8.9 Summary 67 Exercises 8 67 9 PL syntax 69 9.1 Syntactic rules for PL languages 69 9.2 Constructional histories, parse trees 71 9.3 W s have unique parse trees! 73 9.4 Main connectives, subformulas, scope 74 9.5 Bracketing styles 76 9.6 Summary 76 Exercises 9 77
iii 10 PL semantics 78 10.1 Interpreting w s 78 10.2 Languages and translation 80 10.3 Atomic w s are true or false 81 10.4 Truth values 82 10.5 Truth tables for the connectives 83 10.6 Evaluating molecular w s: two examples 84 10.7 Uniqueness and bivalence 85 10.8 Short working 86 10.9 Aside: the very idea of a formalized language 89 10.10 Summary 89 Exercises 10 90 11 P s, Q s, s, s and form again 91 11.1 Styles of variable: object languages and metalanguages 91 11.2 Basic quotation conventions 92 11.3 To Quine-quote or not to Quine-quote 95 11.4 Why Greek-letter variables? 96 11.5 The idea of form, again 97 11.6 Summary 98 Exercises 11 98 12 Truth functions 99 12.1 Truth-functional vs other connectives 99 12.2 Functions and truth functions 100 12.3 Truth tables for w s 101 12.4 Possible valuations 105 12.5 Summary 107 Exercises 12 107 13 Expressive adequacy 108 13.1 Exclusive disjunction 108 13.2 Another example: expressing the dollar truth function 109 13.3 Expressive adequacy defined 110 13.4 Some more adequacy results 111 13.5 Summary 112 Exercises 13 113 14 Tautologies 114 14.1 Tautologies and contradictions 114 14.2 Generalizing examples of tautologies 116 14.3 Tautologies, necessity, and form 117 14.4 Tautologies as analytically true 119 14.5 Summary 119 Exercises 14 120 15 Tautological entailment 121
iv 15.1 Three introductory examples 121 15.2 Tautological entailment defined 123 15.3 Tautological validity and logical validity 124 15.4 Brute-force truth-table testing 125 15.5 More examples 126 15.6 Extending the notion of tautological entailment 128 15.7 Tautological consistency and tautological validity 129 15.8 Summary 130 Exercises 15 131 16 More about tautological entailment 132 16.1 A more e cient test? 132 16.2 Truth-table testing and the counterexample method 133 16.3 and 6 134 16.4 Generalizing examples of tautological entailment 135 16.5 Tautological entailment and form 136 16.6 Explosion! 137 16.7 A slightly di erent definition of tautological entailment 137 16.8 Tautological equivalence as two-way entailment 139 16.9 Summary 140 Exercises 16 141 17 The truth-functional conditional 142 17.1 Some arguments involving conditionals 142 17.2 Four basic principles 143 17.3 Introducing the truth-functional conditional 144 17.4 Ways in which! is conditional-like 145 17.5 Only if (and the biconditional) 148 17.6 Extended PL syntax and semantics, o cially 150 17.7! versus and 6 152 17.8 Summary 153 Exercises 17 154 18 If s and! s 155 18.1 Types of conditional 155 18.2 Simple conditionals as truth-functional: for 156 18.3 Another kind of case where if is truth-functional 158 18.4 Simple conditionals as truth-functional: against 159 18.5 Three responses 160 18.6 Adopting the material conditional 161 18.7 Summary 163 Exercises 18 163 Interlude: Why natural deduction? 164 19 PL proofs: conjunction and negation 168 19.1 Rules for conjunction 168
v 19.2 Rules for negation 170 19.3 A double negation rule 173 19.4 Availability 175 19.5 Proof discovery 177 19.6 Explosion again 178 19.7 Summary 180 Exercises 19 181 20 PL proofs: disjunction 182 20.1 The iteration rule 182 20.2 Introducing and eliminating disjunctions 183 20.3 More examples 187 20.4 Liberalizing the elimination rule 189 20.5 Reviewing the disjunction rules 192 20.6 Theorems 193 20.7 Excluded middle and double negation 195 20.8 Summary 196 Exercises 20 196 21 PL proofs: conditionals 197 21.1 Rules for the conditional 197 21.2 More proofs with conditionals 200 21.3 The material conditional again 203 21.4 Two more examples 203 21.5 Summary 205 Exercises 21 205 22 PL proofs: metatheory 206 22.1 Metatheory 206 22.2 Putting everything together 207 22.3 Vacuous discharge 210 22.4 Generalizing proofs 212 22.5 and ` 212 22.6 Soundness and completeness 214 22.7 How to prove soundness 215 22.8 How to prove completeness 217 22.9 Summary 219 Exercises 22 219 Interlude: Inferentialism, intuitionism 220