Rules of Inference
From Logic Wiki
| 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)
|
