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Chapter 07 Test C
Copyright Cengage Learning. Powered by Cognero.
Page 1
INSTRUCTIONS: Select the conclusion that follows in a single step from the given premises.
1. Given the following premises:
1. C ⊃ (∼L ∨ ∼N)
2. (C • L) ⊃ ∼N
3. N
a. ∼(C • L) 2, 3, MT
b. (C ⊃ ∼L) ∨ ∼N 1, Assoc
c. (C ⊃ ∼N) • (L ⊃ ∼N) 2, Dist
d. C ⊃ ∼N 2, Simp
e. C ⊃ ∼(L • N) 1, DM
2. Given the following premises:
2. ∼A ⊃ (R • M)
3. ∼R • ∼M
a. D ⊃ ∼A 1, Taut
b. D ⊃ A 1, DN
c. D ⊃ (R • M) 1, 2, HS
d. ∼∼A 2, 3, MT
e. ∼(R • M) 3, DM
3. Given the following premises:
1. P ⊃ L
2. ∼(J • O)
3. (L ⊃ A) ⊃ (J • O)
a. L ⊃ P 1, Com
b. ∼J • ∼O 2, DM
c. P ⊃ A 1, 3, HS
d. ∼(L ⊃ A) 2, 3, MT
e. ∼J 2, Simp
4. Given the following premises:
1. E ⊃ (B • J)
2. (J • B) ⊃ ∼L
3. L
a. E ⊃ ∼L 1, 2, HS
b. ∼(J • B) 2, 3, MT
c. (B • J) ⊃ ∼L 2, Com
d. J 2, Simp
e. (E ⊃ B) • (E ⊃ J) 1, Dist
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Chapter 07 Test C
Copyright Cengage Learning. Powered by Cognero.
Page 2
5. Given the following premises:
1. F ∨ S
2. ∼S
3. (S ⊃ W) • (F ⊃ N)
a. F 1, 2, DS
b. S ⊃ W 3, Simp
c. ∼F ⊃ S 1, Impl
d. F ⊃ N 3, Simp
e. W ∨ N 1, 3, CD
6. Given the following premises:
1. (E ⊃ K) ∨ W
2. ∼W
3. W ∨ ∼(Q ⊃ E)
a. E ⊃ K 1, 2, DS
b. Q ⊃ K 1, 3, HS
c. ∼(Q ⊃ E) 2, 3, DS
d. E ⊃ (K ∨ W) 1, Assoc
e. W ∨ (∼Q ⊃ ∼E) 3, DM
7. Given the following premises:
1. E
2. R ⊃ ∼E
3. N ⊃ (∼C ⊃ R)
a. ∼R 1, 2, MT
b. E • H 1, Add
c. ∼C ⊃ ∼E 2, 3, HS
d. E ⊃ ∼R 2, Trans
e. (N • ∼C) ⊃ R 3, Exp
8. Given the following premises:
1. ∼(G • F)
2. ∼F ⊃ H
3. (G ⊃ ∼F) • (∼F ⊃ G)
a. ∼F ⊃ G 3, Simp
b. G ⊃ H 2, 3, HS
c. F ∨ H 2, Impl
d. G ≡ ∼F 3, Equiv
e. ∼G 1, Simp
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Chapter 07 Test C
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9. Given the following premises:
2. G
3. ∼T
a. A 1, 2, MP
b. G • ∼T 2, 3, Conj
c. G ⊃ A 1, 3, DS
d. G ⊃ (A ∨ T) 1, Assoc
e. G ⊃ (A ⊃ T) 1, Exp
10. Given the following premises:
1. Q ⊃ (∼N ∨ ∼N)
2. ∼N ⊃ ∼∼P
3. P ⊃ ∼G
a. ∼N ⊃ P 2, DN
b. Q ⊃ ∼∼P 1, 2, HS
c. N ∨ P 2, Impl
d. ∼N ⊃ ∼G 2, 3, HS
e. G ⊃ ∼P 3, Trans
11. Given the following premises:
1. T ⊃ (G ∨ G)
2. ∼P ⊃ T
3. F ⊃ (B ⊃ ∼P)
a. F ⊃ (P ⊃ ∼B) 3, Trans
b. (F ⊃ B) ⊃ ∼P 3, Assoc
c. F ⊃ (∼B ∨ ∼P) 3, Impl
d. B ⊃ T 2, 3, HS
e. ∼P ⊃ G 1, 2, HS
12. Given the following premises:
1. C ⊃ (H • M)
2. (T ⊃ S) ⊃ C
3. T
a. (C ⊃ H) • M 1, Assoc
b. T ⊃ (S • C) 2, Exp
c. (C ⊃ H) • (C ⊃ M) 1, Dist
d. S 2, 3, MP
e. (T ⊃ S) ⊃ (H • M) 1, 2, HS
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Chapter 07 Test C
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13. Given the following premises:
1. ∼W
2. C ∨ W
3. R ⊃ ∼(C ∨ W)
a. R ⊃ (∼C • ∼W) 3, DM
b. ∼R 2, 3, MT
c. C 1, 2, DS
d. (C ∨ W) ⊃ ∼R 3, Trans
e. ∼C ⊃ W 2, Impl
14. Given the following premises:
1. B
2. ∼R ⊃ K
3. B ⊃ (K ⊃ E)
a. (B ⊃ K) ⊃ E 3, Assoc
b. ∼R ⊃ E 2, 3, HS
c. R ∨ K 2, Impl
d. K ⊃ E 1, 3, MP
e. B • N 1, Add
15. Given the following premises:
1. ∼A ⊃ ∼S
3. ∼Q
a. ∼A 2, 3, MP
b. (E ⊃ ∼Q) ⊃ ∼A 2, Assoc
c. E ⊃ (A ⊃ Q) 2, Trans
d. ∼Q ⊃ ∼S 1, 2, HS
e. A ∨ ∼S 1, Impl
16. Given the following premises:
1. ∼(F • J)
2. ∼F
3. (F • H) ∨ (F • J)
a. F • H 1, 3, DS
b. F • (H ∨ J) 3, Dist
c. F ∨ (H • J) 3, Dist
d. ∼F • ∼J 1, DM
e. F 3, Simp
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Chapter 07 Test C
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Page 5
17. Given the following premises:
1. H ∨ M
2. E ⊃ ∼(H ∨ M)
3. (H ⊃ D) • (M ⊃ O)
a. ∼H ⊃ M 1, Impl
b. ∼E 1, 2, MT
c. H 1, Simp
d. M ⊃ O 3, Simp
e. D ∨ O 1, 3, CD
18. Given the following premises:
1. N ⊃ ∼(S ∨ K)
2. S ∨ K
3. S ⊃ (R • Q)
a. S 2, Simp
b. (S ∨ K) ∨ N 2, Add
c. ∼S ⊃ K 2, Impl
d. ∼N 1, 2, MT
e. (S ⊃ R) ⊃ Q 3, Exp
19. Given the following premises:
1. ∼(∼H • J)
2. K ∨ (∼H • J)
3. (M ∨ M) ⊃ (∼H • J)
a. (K ∨ ∼H) • (K ∨ J) 2, Dist
b. ∼K ⊃ (∼H • J) 2, Impl
c. K 1, 2, DS
d. H ∨ ∼J 1, DM
e. ∼M 1, 3, MT
20. Given the following premises:
1. S ∨ (∼Q ∨ ∼C)
2. (∼Q ∨ ∼C) ⊃ M
3. T ⊃ (Q • C)
a. S ⊃ M 1, 2, HS
b. S ∨ ∼(Q ∨ C) 1, DM
c. (S ∨ ∼Q) ∨ C 1, Assoc
d. ∼Q ∨ (∼C ⊃ M) 2, Assoc
e. (T ⊃ Q) • (T ⊃ C) 3, Dist
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Chapter 07 Test C
Copyright Cengage Learning. Powered by Cognero.
Page 6
INSTRUCTIONS: Use natural deduction to derive the conclusion in each problem.
21. Use an ordinary proof (not conditional or indirect proof):
1. K ∨ (S • N)
2. ∼(K • ∼Q)
3. ∼(N • ∼Q) / Q
22. Use an ordinary proof (not conditional or indirect proof):
1. M ⊃ (R • E)
2. (E ∨ H) ⊃ G / M ⊃ G
23. Use an ordinary proof (not conditional or indirect proof):
1. F ⊃ (J ∨ ∼F)
2. J ⊃ (L ∨ ∼J) / F ⊃ L
24. Use conditional proof:
1. S ⊃ (B ⊃ T)
2. N ⊃ (T ⊃ ∼B) / (S • N) ⊃ ∼B
25. Use indirect proof:
1. (P ∨ F) ⊃ (A ∨ D)
2. A ⊃ (M • ∼P)
3. D ⊃ (C • ∼P) / ∼P
26. Use natural deduction to prove the following logical truth:
(P ⊃ Q) ≡ [P ⊃ (Q ∨ ∼P)]
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