An Introduction to. Formal Logic. Second edition. Peter Smith, February 27, 2019

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1 An Introduction to Formal Logic Second edition Peter Smith February 27, 2019

2 Peter Smith Not for re-posting or re-circulation. Comments and corrections please to ps218 at cam dot ac dot uk

3 1 What is deductive logic? What is an argument? Kinds of evaluation Deduction vs. induction Just a few more examples Generalizing Summary 7 Exercises Validity and soundness Validity defined Consistency, validity, and equivalence Validity, truth, and the invalidity principle Inferences and arguments Valid vs true What s the use of deduction? An illuminating circle? Summary 18 Exercises Forms of inference More forms of inference Four basic points about the use of schemas Arguments can instantiate many patterns Summary 26 Exercises Proofs Proofs: first examples Fully annotated proofs Glimpsing an ideal Deductively cogent multi-step arguments Indirect arguments Summary 35 Exercises 4 36

4 ii 5 The counterexample method But you might as well argue The counterexample method, more carefully A quantifier shift fallacy Summary 41 Exercises Logical validity Topic neutrality Logical validity, at last Logical necessity The boundaries of logical validity? Definitions of validity as rational reconstructions Summary 48 Exercises Propositions and forms Types vs tokens Sense vs tone Are propositions sentences? Are propositions truth-relevant contents? Why we can be indecisive Forms of inference again Summary 54 Interlude: From informal to formal logic 55 8 Three connectives Two simple arguments And Or Not Scope Formalization The design brief for PL languages One PL language Summary 67 Exercises PL syntax Syntactic rules for PL languages Constructional histories, parse trees W s have unique parse trees! Main connectives, subformulas, scope Bracketing styles Summary 76 Exercises 9 77

5 iii 10 PL semantics Interpreting w s Languages and translation Atomic w s are true or false Truth values Truth tables for the connectives Evaluating molecular w s: two examples Uniqueness and bivalence Short working Aside: the very idea of a formalized language Summary 89 Exercises P s, Q s, s, s and form again Styles of variable: object languages and metalanguages Basic quotation conventions To Quine-quote or not to Quine-quote Why Greek-letter variables? The idea of form, again Summary 98 Exercises Truth functions Truth-functional vs other connectives Functions and truth functions Truth tables for w s Possible valuations Summary 107 Exercises Expressive adequacy Exclusive disjunction Another example: expressing the dollar truth function Expressive adequacy defined Some more adequacy results Summary 112 Exercises Tautologies Tautologies and contradictions Generalizing examples of tautologies Tautologies, necessity, and form Tautologies as analytically true Summary 119 Exercises Tautological entailment 121

6 iv 15.1 Three introductory examples Tautological entailment defined Tautological validity and logical validity Brute-force truth-table testing More examples Extending the notion of tautological entailment Tautological consistency and tautological validity Summary 130 Exercises More about tautological entailment A more e cient test? Truth-table testing and the counterexample method and Generalizing examples of tautological entailment Tautological entailment and form Explosion! A slightly di erent definition of tautological entailment Tautological equivalence as two-way entailment Summary 140 Exercises The truth-functional conditional Some arguments involving conditionals Four basic principles Introducing the truth-functional conditional Ways in which! is conditional-like Only if (and the biconditional) Extended PL syntax and semantics, o cially ! versus and Summary 153 Exercises If s and! s Types of conditional Simple conditionals as truth-functional: for Another kind of case where if is truth-functional Simple conditionals as truth-functional: against Three responses Adopting the material conditional Summary 163 Exercises Interlude: Why natural deduction? PL proofs: conjunction and negation Rules for conjunction 168

7 v 19.2 Rules for negation A double negation rule Availability Proof discovery Explosion again Summary 180 Exercises PL proofs: disjunction The iteration rule Introducing and eliminating disjunctions More examples Liberalizing the elimination rule Reviewing the disjunction rules Theorems Excluded middle and double negation Summary 196 Exercises PL proofs: conditionals Rules for the conditional More proofs with conditionals The material conditional again Two more examples Summary 205 Exercises PL proofs: metatheory Metatheory Putting everything together Vacuous discharge Generalizing proofs and ` Soundness and completeness How to prove soundness How to prove completeness Summary 219 Exercises Interlude: Inferentialism, intuitionism 220

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