1. Chapter 4 Natural Deduction Different ways of formulating a logical system: Axiomatic and natural deduction Mental logic is natural deductive Key feature: Allowing assumptions without deductive theorem Purely syntactic: Proof-theoretic approach

2. Format Categorical proof: No assumption Hypothetical proof: With assumption Format of a proof: 1. a formula (Premise 1 or assumption) … n. a formula (Premise n) Show: a formula (the conclusion)

3. Inference Rules Simple rule: Copy a formula established previously For each truth connective, there are a pair of rules: introduction and elimination A rule is always applied to the main connective in a formula

4. Conjunction Rules Conjunction Elimination (&E): A & B A(&E) Ex. 1. (p & q) & ¬r(Premise) 2. Show q(Conclusion) 3. p & q(1, &E) 4. q(3, &E, mark 2)

5. Conjunction Introduction (&I) A B ____________ A & B Ex. 1. p & q(premise) 2. show q & p(conclusion)

6. Negation Rules Negation Introduction (NI): A / ¬¬A Negation Elimination (NE): ¬¬A / A Ex. 1. ¬¬(p & ¬¬¬q) (Premise) 2. Show ¬¬p & ¬q (this proof has 8 steps to complete)

7. Indirect Proof 1. Show ¬A 2. A(AIP, assumption for indirect proof) | … n. | B n+1| ¬B Mark 1 by discharging 2.

8. Examples (p & ¬p) / q show ¬ ¬q Assume ¬q (AIP)