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
S v S Taut
S S Taut
~~S DN
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)
(H v J) v B Com
B v (J v H) Com
(B v H) v J Assoc
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
6. [F (V F)] [(Z G) v (G T)]
(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