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 1.F > G 2.F v G / G 3. ~F > G IMP 2 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 a candidate wins Ohio and Pennsylvania, then she’ll win the election : : If a candidate wins Ohio, then if she wins Pennsylvania, then she’ll win the election

5. Material Equivalence EQ (p  q) : : [( p > q). (q > p)] “if and only if” means “necessary and sufficient condition for” Biconditional interpretation ( 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. 1.P > Q 2.R > (S. T) 3. ~R > ~Q 4.S > ( T > P) / P  R 5. Q > R TRANS 3 6. P > R HS 1,5 (P > R). (R >P) 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. 7. 3 III, 24 1.(M.N) v ( O. P) 2.(N v O) > ~P / N [p v (q.r)] :: [(p v q). (p v r)] (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