Philosophy Rules

p→ q

p

∴ q

Rule 1. Modus Ponens (MP)

p → q

~q

∴ ~p

Rule 2. Modus Tolles (MT)

p → q

q → r

∴ p→ r

Rule 3. Hypothetical Syllogism (HS)

p v q

~p

∴ q

Rule 4. Disjunctive Syllogism (DS)

q v p

~q

∴ p

Rule 4. Disjunctive Syllogism (DS)

p v q

p → r

q → s

∴ r v s

Rule 5. Constructive Dilemma (CD)

p • q

∴p

Rule 6. Simplification (Simp)

q • p

∴q

Rule 6. Simplification (Simp)

p

q

∴ p • q

Rule 7. Conjunction (Conj)

p

∴ p v q

Rule 8. Addition (Add)

q

∴ q v p

Rule 8. Addition (Add)

p : : ~~p

Rule 9. Double negation (DN)

If p is a statment in a proof, we can replace it with ~~p.

p v q : : q v p / p • q :: q • p

Rule 10. Commutation (Com)

p v (q v r) :: (p v q) v r / (Can also be •)

Rule 11. Association (As)

~(p v q) :: ~p • ~q (Can be reversed)

Rule 12. DeMorgan’s law (DeM)

p → q :: ~q → ~p

Rule 13. Contraposition (Cont)

p • (q v r) :: (p • q) v (p • r)

(v & • can be inverted)

Rule 14. Distribution (Dist)

(p • q) → r :: p → (q → r)

Rule 15. Exportation (Exp)

p :: p • p / p :: p v p

Rule 16. Redundancy (Re)

p ←→ q :: (p → q) • (q → p)

p ←→ q :: (p • q) v (~q • ~p)

Rule 17. Material Equivalence (ME)

p → q :: ~p v q

Rule 18. Material Implication (MI)