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.

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 premise2. ~M v ~G___ 1, ad3. ~(M . G)___ 2, dm

1. ~(H v K) premise2. ~ H . ~ K 1. dm

3. ~K ___ 2 cm, sm

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.

Material Implication IMP

(p v q) :: (~p > q) “or” means the same thing as “if not”

1. ~A premise2. (M > L) v A premise3. ~(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 IMPA > B DN

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

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.

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

Tautology TAUT

(p v p) : : p

(p . p) : : p

Eliminates redundancy

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

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