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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