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

~ ( 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

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

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

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.

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)]