1. 7.4 Rules of Replacement II Trans, Impl, Equiv, Exp, Taut

2. 14. Transposition (Trans) (P  Q) :: (~Q  ~P) If Renée is a Californian then she is from the west coast. If she is not from the west coast, then she is not a Californian. P  Q::~Q  ~P TTTTFTF TFFTTFF FTTTFTT FTFTTTT

3. 14. Trans 1. ~(T v R)  S 2. ~S  (T v R)1, trans 1. H v (K  ~D) 2. H v (D  ~K)1, trans

4. 14. Trans Trans can be used to set up HS. 1. A  B 2. ~C  ~B 3. B  C2, Trans 4. A  C1,3 HS

5. 15. Impl 15. Material Implication (Impl) (P  Q) :: (~P v Q) “If you bother me then I will punch you in the nose.” “Either you stop bothering me or I will punch you in the nose.”

6. 15. Impl Impl can be used to set up HS. 1. ~A v B 2. ~B v C 3. A  B1, Impl 4. B  C2, Impl 5. A  C3,4 HS

7. 15. Impl 1. (G  R)  (H v B) 2. G v ~H/ R v B 3. (G  R)  (~H  B) 1, Impl 4. R v B1,3, CD

8. 16. Equiv 16. Material Equivalence (Equiv) (P  Q) :: [(P  Q)  (Q  P)] “P iff Q” :: “if P then Q, and if Q then P” (P  Q) :: [(P  Q) v (~Q  ~P)] “P iff Q” :: “P and Q are both true, or they are both false”

9. 17. Exp 17. Exportation (Exp) [(P  Q)  R] :: [P  (Q  R)] “If we have P, then if we have Q we have R” “If we have both P and Q, then we have R”

10. 17. Exp Exportation can be used to set up MT. 1. A  (B  C) 2. ~C 3. (A  B)  C1, Exp 4. ~(A  B) 2,3 MT

11. 18. Taut 18. Tautology (Taut) P :: P v PP :: P  P

12. 14. Trans(P  Q) :: (~Q  ~P) 15. Impl(P  Q) :: (~P v Q) 16. Equiv(P  Q) :: [(P  Q)  (Q  P)] 17. Exp[(P  Q)  R] :: [P  (Q  R)] 18. TautP :: P v P P :: P  P

13. 7.4.1 p. 52 Provide logically equivalent statements using the rules of replacement. Do 1-10. Do the evens if you were born on an even numbered day. Do odds if you were born on an odd numbered day.

14. 1. S

15. S v S Taut S  STaut ~~SDN

16. 2. A  (F  B)

17. ~~A  (F  B)DN A  (B  F)Com [A  (F  B)]  [(F  B)  A] Equiv (A v A)  (F  B) Taut

18. 3. B v (H v J)

19. (H v J) v B Com B v (J v H)Com (B v H) v JAssoc

20. 4. J  (K  P)

21. ~J v (K  P) Impl ~(K  P)  ~J Trans J  [(K  P)  (P  K)]Equiv J  [(K  P) v (~P  ~K)] Equiv

22. 5. J  (K  P)

23. 6. [F  (V  F)]  [(Z  G) v (G  T)]

24. 7. R  [(W  V)  Q]

25. 8. S v (P v A)

26. 9. ~(A  G)

27. 10. ~(A  G) v ~P

28. (J  R)  H (R  H)  M ~(P v ~J)/ M  ~P ___________1, Exp ___________2, 4, HS ___________3, DM ___________6, DN ___________7, Simp ___________7, Com ___________9, Simp ___________5, 10, MP ___________8, 11, Conj