Rules of Inference

Order	Rule Name	Formula
1	Modus Ponens (M.P.)	p → q p .: q
2	Modus Tollens (M.T.)	p → q ~ q .: ~ p
3	Hypothetical Syllogism (H.S.)	p → q q → r .: p → r
4	Disjunctive Syllogism (D.S.)	p v q ~ p .: q
5	Constructive Dilemma (C.D.	(p → q) ∙ (r → s) p v r .: q v s1
6	Absorption (Abs.)	p → q .: p → (p∙q)
7	Simplification (Simp.)	p∙q .: p
8	Conjunction (Conj.)	p q .: p∙q
9	Addition (Add.)	p .: p v q
10	De Morgan’s Theorem (De M.)	~ (p∙q) ≡ (~ p v ~ q) ~ (p v q) ≡ (~ p∙~ q)
11	Commutation (Com.)	(p v q) ≡ (q v p) (p∙q) ≡ (q∙p)
12	Association (Assoc.)	[p v (q v r)] [(p v q) v r] [p∙ (q∙r)] [(p∙q) ∙r]
13	Distribution (Dist)	[p∙(q v r)] ≡ [(p∙q) v (p∙r)] [p v (q∙r)] ≡ [(p v q) ∙ (p v r)]
14	Double Negation (D.N.)	p ≡ ~ ~ p
15	Transposition (Trans.)	(p → q) ≡ (~ q → ~ p)
16	Material Implication (M. Imp.)	(p → q) ≡ (~ p v q)
17	Material Equivalence (M. Equiv.)	(p≡q) ≡ [(p → q) ∙ (q → p)] (p≡q) [(p∙q) v (~ p ∙ ~ q)]
18	Exportation (Exp.)	[(p∙q) → r] ≡ [p → (q → r)]
19	Tautology (Taut.)	p ≡ (p v p) p ≡ (p∙p)