1. DeMorgan’s Rule DM ~(p v q) :: (~p. ~q)“neither…nor…” is the same as “not the one and not the other” ~( p. q) :: (~p v ~q) “not both…” is the same as “either not this one or not that one.” 1.~M premise 2.~M v ~G___ 1, ad 3. ~(M. G)___2, dm 1.~(H v K) premise 2.~ H. ~ K 1. dm 3. ~K ___ 2 cm, sm

2. Transposition TR (p > q) :: (~q > ~p) Contraposition, but in the context of propositional logic All Popes are Catholics so All non-Catholics are non-Popes. If he’s the Pope, he’s Catholic, so if he’s not Catholic, he’s not the Pope.

3. Material Implication IMP (p v q) :: (~p > q) “or” means the same thing as “if not” 1.~A premise 2.(M > L) v A premise 3.~(M > L) > A_2_IMP_ 4. ~A > (M > L)_3_TR_ 5. M > L_1,4_mp_ When you change a “v” to a “>” or vice versa, add a tilde to the expression on the left A > B ~A v B IMP ~A v B ~~A > B IMP A > B DN

4. Distribution DIST [ p v (q. r)] : : [(p v q). (p v r)] [p. (q v r)] : : [(p. q) v (p. r)] “p” is being distributed through a disjunction or a conjunction

5. Material Equivalence EQ (p ≡ q) : : [(p > q). (q > p)] (p ≡ q) : : [(p. q) v (~p. ~q) Biconditional: p and q are necessary and sufficient conditions for each other: p implies q and q implies p. They have the same truth values: either both are true or both are false. Either both or neither.

6. Exportation EXP [p > (q > r)] : : [(p. q) > r] If p is true, then if q is, so is r if p and q are both true, then so is r

7. Tautology TAUT (p v p) : : p (p. p) : : p Eliminates redundancy

8. Rules of inference (8) MP p > q / p // q MT p > q / ~q // ~p HS p > q / q > r // p > r DS p v q / ~p // q SM p. q // p CN p / q // p. q AD p // p v q CD (p > q). (r > s) / p v r // q v s

9. Rules of Equivalence/ Replacement (10) DN p :: ~~p CM (p. q) :: (q. p) (p v q) :: (q v p) AS ((p. q). r) :: (p. ( q. r)) ((p v q) v r) :: (p v (q v r)) DM ~(p v q) :: (~p. ~q) ~(p.q) :: (~p v ~q) DIST (p v (q. r)) :: ((p v q). (p v r)) (p. (q v r)) :: ((p. q) v (p. r)) TRAN (p > q) :: (~q > ~p) IMP (p v q) :: (~p > q) EQ (p ≡ q) :: ((p > q). (q > p)) (p ≡ q) :: ((p. q) v (~p. ~q)) EXP (p > (q > r)) :: ((p. q) > r) TAUT (p v p) :: p (p. p) :: p