Rules of Implication Rules of Replacement Modus Ponens (MP) DeMorgan’s rule (DM) p q q) (∼p ⩷ ⊃ ∼(p ∙ ∨ ∼q) p_____ q) ⩷ (∼p ∙ ∼(p ∨ ∼q) q Commutativity (Com) Modus Tollens (MT) (p∨q) (q∨p) ⩷ p q ⊃ (p q) (q p) ⩷ ∙ ∙ ∼q____ Associativity (Assoc) ∼p [p (q r)] [(p q) r] ⩷ ∨ ∨ ∨ ∨ Pure Hypothetical Syllogism (HS) [p (q r)] [(p q) r] ⩷ ∙ ∙ ∙ ∙ p q ⊃ Distribution (Dist) q r_ ⊃ [p (q r)] [(p q) (p r)] ⩷ ∙ ∙ ∙ ∨ ∨ p r ⊃ [p (q r)] [(p q) (p r)] ⩷ ∨ ∙ ∨ ∙ ∨ Disjunctive Syllogism (DS) Double Negation (DN) p q ⋁ p ⩷ ∼∼p ∼p___ q Constructive dilemma (CD) (p q) (r s) ⊃ ⋅ ⊃ p r____________ ⋁ q s ⋁ Simplification (Simp) p ∙ q_ p Conjunction (Conj) p q_ p q ∙ Addition (Add) p_____ p q ⋁
Rules of Implication Modus Ponens (MP) p ⊃ q p_____ q Modus Tollens (MT) p ⊃ q ∼q____ ∼p Pure Hypothetical Syllogism (HS) p ⊃ q q ⊃ r_ p ⊃ r Disjunctive Syllogism (DS) p ⋁ q ∼p___ q Constructive dilemma (CD) (p ⊃ q) ⋅ (r ⊃ s) p ⋁ r____________ q ⋁ s Simplification (Simp) p ∙ q_ p Conjunction (Conj) p q_ p ∙ q Addition (Add) p_____ p ⋁ q Rules of Replacement DeMorgan’s rule (DM) ∼(p ∙ q) ⩷ (∼p ∨ ∼q) ∼(p ∨ q) ⩷ (∼p ∙ ∼q) Commutativity (Com) (p∨q) ⩷ (q∨p) (p ∙ q) ⩷ (q ∙ p) Associativity (Assoc) [p ∨ (q ∨ r)] ⩷ [(p ∨ q) ∨ r] [p ∙ (q ∙ r)] ⩷ [(p ∙ q) ∙ r] Distribution (Dist) [p ∙ (q ∨ r)] ⩷ [(p ∙ q) ∨ (p ∙ r)] [p ∨ (q ∙ r)] ⩷ [(p ∨ q) ∙ (p ∨ r)] Double Negation (DN) p ⩷ ∼∼p