# Deductive Logic/Equivalent Schemata

## Equivalent Schemata

editA number of schemata can be proven to be equivalent. These are useful for transforming an argument into a more convenient, yet equivalent form.

Several valid equivalent schemata will be introduced, including the name and forms for each. Recall that the ≡ symbol represents equivalence.

**Double Negative**: ‘p’ ≡ ‘¬ (¬ p)’

**Association**: ‘p∨(q∨r’) ≡ ‘(p∨q)∨r’ , also: ‘p•(q•r)’ ≡ ‘(p•q)•r’

**Commutation**: ‘p∨q’ ≡ ‘q∨p’ , also: ‘p•q’ ≡ ‘q•p’

**De Morgan’s laws**: ‘¬ (p∨q)’ ≡ ‘¬ p • ¬ q’ , also: ‘¬ (p•q)’ ≡ ‘¬ p ∨¬ q’

**Distribution**: ‘p•(q∨r)’ ≡ ‘(p•q) ∨(p•r)’ , also: ‘p ∨(q•r)’ ≡ ‘(p∨q) • (p∨r)’

**Transposition (Contraposition):** ‘p → q’ ≡ ‘¬ q → ¬ p’ ("twist it around and not it and it will hold")

**Implication**: ‘p → q’ ≡ ‘¬ p∨q’ , also: ‘p → q’ ≡ ‘¬(p• ¬ q)’

**Idempotence**: ‘p’ ≡ ‘p•p’, also: ‘p’ ≡ ‘p∨p’

**Exportation**: ‘(p•q) → r’ ≡ ‘p → (q → r)’

Biconditional: ‘p≡q’ ≡ ‘(p → q) • (q → p)’, also: ‘p≡q’ ≡ ‘(p • q) ∨(¬ p • ¬ q)’

**Law of the Excluded middle**: ‘p ∨¬ p’ ≡ true