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

7.4 Rules of Replacement II

Trans, Impl, Equiv, Exp, Taut

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

T T T T F T F

T F F T T F F

F T T T F T T

F T F T T T T

14. Trans

1. ~(T v R) S2. ~S (T v R) 1, trans

1. H v (K ~D)2. H v (D ~K) 1, trans

14. Trans

Trans can be used to set up HS.

1. A B2. ~C ~B3. B C 2, Trans4. A C 1,3 HS

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.”

15. Impl

Impl can be used to set up HS.

1. ~A v B2. ~B v C3. A B 1, Impl4. B C 2, Impl5. A C 3,4 HS

15. Impl

1. (G R) (H v B)2. G v ~H / R v B3. (G R) (~H B) 1,

Impl4. R v B 1,3, CD

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”

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”

17. Exp

Exportation can be used to set up MT.

1. A (B C)2. ~C3. (A B) C 1, Exp4. ~(A B) 2,3 MT

18. Taut

18. Tautology (Taut)

P :: P v P P :: P P

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. Taut P :: P v PP :: P P

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.

1. S

1. S

S v S Taut

S S Taut

~~S DN

2. A (F B)

2. A (F B)

~~A (F B) DN

A (B F) Com

[A (F B)] [(F B) A] Equiv

(A v A) (F B) Taut

3. B v (H v J)

3. B v (H v J)

(H v J) v B Com

B v (J v H) Com

(B v H) v J Assoc

4. J (K P)

4. J (K P)

~J v (K P) Impl

~(K P) ~J Trans

J [(K P) (P K)] Equiv

J [(K P) v (~P ~K)] Equiv

5. J (K P)

6. [F (V F)] [(Z G) v (G T)]

7. R [(W V) Q]

8. S v (P v A)

9. ~(A G)

10. ~(A G) v ~P

(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