Deductive Logic/Equivalent Schemata

Equivalent Schemata
A 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