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