1. Double Negation
p :: ~ ~p
1.A > ~ (B . C)
2. B . C
~ ~ ( H v K)
~ H
3. ~ ~ (B . C) 2 DN
4. ~A ,3 MT
3. H v K 1 DN
4. K ,3 DS

2. ~ ( N v W)
:: (~N . ~W)
Neither Nader nor Whatshisname
will be elected :: Nader won’t be
and Whatshisname won’t be.
~ ( B . K)
:: (~ B v ~K)
Not both Bush and Kerry will win:
Either Bush won’t be or Kerry
won’t be.
deMorgan’s Rule DM
~ (p v q) :: (~p . ~q)
~ (p . q) :: (~ p v ~q)
1. ~( N v W)
2. B > N / ~B
3. ~ N . ~ W 1 DM
4. ~N 3 SIMP
5. ~B MT 2,4

3. COMM Commutation
(p . q) : : (q . p)
(p v q) : : (q v p)
(J > N) v (C v D) ~C
~D / J >N
~ C . ~ D CONJ 2,3,
~ (C v D) DM 4
J > N COMM, DS 1,5
6. (C v D) v (J >N) COMM, 1
7. J >N DS 5, 6

4. Association ASSOC
[ p . (q . r)] : : [ (p . q) . r]
[ p v (q v r)] : : [ (p v q) v r]
1. A . (B . C) / C
2. (A . B) . C ASSOC 1
3. C . (A . B) COMM 2
4. C SIMP 3

5. Distribution
[p v (q . r)] : : [(p v q) . (p v r)]
We either eat at Seacobeck or we go downtown and eat at
Bangkok Café : :
Either we eat at Seacobeck or go downtown and either we
eat at Seacobeck or we eat at Bangkok Café.
[ p. (q v r)] : : [ p . q) v (p . r)]
We’re going downtown and either we’ll eat at Bangkok
or at Foodes : :
We’re going downtown and eating at Bangkok or else we’re
going downtown and eating at Foodes.

6. Rules of Replacement
Double Negation DN
p :: ~ ~p
deMorgan’s Rule DM
~ (p v q) :: (~p . ~q)
~ (p . q) :: (~ p v ~q)
Commutation COMM
(p . q) : : (q . p)
(p v q) : : (q v p)
Association ASSOC
[ p . (q . r)] : : [ (p . q) . r]
[ p v (q v r)] : : [ (p v q) v r]
Distribution DIST
[p v (q . r)] : : [(p v q) . (p v r)]
[ p. (q v r)] : : [ p . q) v (p . r)]