Transposition (p > q) : : ( ~ q > ~ p) Negate both statements

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Presentation transcript:

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

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

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

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

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

(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

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

[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