1. Transposition
(p > q) : : ( ~ q > ~ p)
Negate both statements
when switching order of
antecedent and consequent
If the car starts, there’s gas in the tank ::
If there’s no gas in the tank, the car won’t start

2. Implication
(p > q) : : (~ p v q)
Lets you replace a “v” with a “>” and vice versa, as long
as you negate the first statement.
If you want to get better at violin you have to practice : :
either you don’t want to get better at violin or else you’ll
practice. V > P ~V v P
Either you register by Oct 4 or else you can’t vote : :
If you don’t register by Oct 4, you can’t vote
R v ~C ~R > ~C

3. Tautology
(p v p) : : p
(p . p) : : p
Eliminates or introduces redundancies
F > G
F v G / G
3. ~F > G IMP 1
4. ~G > F TRANS 3
5. ~G > G HS 4,1
6. G v G IMP 5
7. G TAUT 6

4. Exportation
[(p . q) > r] : : [( p > (q > r)]
If two conditions together imply a statement, then
if the one is true, then the other implies the statement,
and vice versa.
If Kerry wins Ohio and Pennsylvania, then he’ll win the
election : :
If Kerry wins Ohio, then if he wins Pennsylvania, then he’ll
win the election

5. Material Equivalence EQ
(p  q) : : [( p > q) . (q > p)]
Biconditional interpretation
“if and only if” means “necessary and sufficient condition for”
( p  q) : : [( p . q) v (~p . ~q)]
Equivalency interpretation
They are the same : : Either they are both
true or they are both false
They are the same : : Either both or neither

6. (P > R) . (R >P)
P > Q
R > (S . T)
~R > ~Q
S > ( T > P) / P  R
5. Q > R TRANS 5
6. P > R HS 1,5
7. (S . T) > P EXP 4
8. R > P HS 2,7
9. (P > R) . (R > P) CONJ 6, 8
10. P  R EQ 9

7. 1. A / ~B > A
2. A v B ADD 1
3. B v A COMM 2
4. ~B > A IMP 3

8. [p v (q .r)] :: [(p v q) . (p v r)]
7. 3 III, 24
(M .N) v ( O . P)
(N v O) > ~P / N
(M . N) v ( O . P)
p v (q . r)
3. [(M . N) v O] . [( M . N) v P] DIST 1
4. (M . N) v O SIMP 3
5. O v ( M . N) COMM 4
6. (O v M) . (O v N) DIST 5
7. O v N COMM, SIMP 6
8. N v O COMM 7
9. ~ P MP 2,8
10 . (M . N) v P COMM, SIMP 3
11. M . N COMM, DS 9, 10
12. N COMM, SIMP 11