Prof. Harasimchuk Phil Dept / Fall 2016 Formal Proofs: Rules of Inference Handout Ch 7 Natural Deduction in Propositional Logic (Hurley) Operator • ∨ ⊃ Breaks Up Operator Creates New Operator SIMP p • q p SIMQ p • q q CONJ p q p • q DS p ∨ q ∼p q DSQ p ∨ q ADD MP p ⊃ q p q ∼q p p ∨ q p MT p ⊃ q ∼q ∼p HS p ⊃ q q ⊃ r p ⊃ r CD (p ⊃ q) • (r ⊃ s) p ∨ r q ∨ s ABS p ⊃ q p ⊃ (p • q) pg. 1 Formal Proofs: Rules of Replacement Handout Prof. Harasimchuk Phil Dept / Fall 2016 Ch 7 Natural Deduction in Propositional Logic (Hurley) COM DIST p•q ↔ q•p p • (q ∨ r) ↔ (p • q) ∨ (p • r) p∨q ↔ q∨p p ∨ (q • r) ↔ (p ∨ q) • (p ∨ r) p≡q ↔ (p ⊃ q) • (q ⊃ p) p≡q ↔ (p • q) ∨ (∼p • ∼q) TRANS p⊃q EQUIV ↔ ∼q ⊃ ∼p TAUT p ↔ p•p p ↔ p∨p CP DN p ↔ ∼∼p 1) 2) /p⊃q i) p MI p⊃q ↔ ACP ii) ∼p ∨ q iii) iv) q DeM ∼(p • q) ↔ ∼p ∨ ∼q ∼(p ∨ q) ↔ ∼p • ∼q v) p⊃q IP EXP 1) (p • q) ⊃ r ↔ p ⊃ (q ⊃ r) 2) /p i) ASSOC p • (q • r) i-iv CP ∼p AIP iv) q • ∼q CONJ ii) ↔ (p • q) • r iii) p ∨ (q ∨ r) ↔ (p ∨ q) ∨ r v) p i-iv IP pg. 2