Question 1

Given the following premises:
1)∼R ∨ ∼R
2)R ∨ (∼J • ∼H)
3)∼R ⊃ (H • B)

Accepted Answer

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 
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

Question 2

Given the following premises:
1)Q ⊃ (∼N ∨ ∼N)
2)∼N ⊃ ∼∼P
3)P ⊃ ∼G

Accepted Answer

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 
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

Question 3

Given the following premises:
1)R • ∼S
2)R ⊃ ∼(S • ∼F)
3)∼S ⊃ (F • N)

Accepted Answer

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 
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

Question 4

Use an ordinary proof (not conditional or indirect proof):
1.A ⊃ (Q ∨ R)
2.(R • Q) ⊃ B
3.A • ∼B
/ R ≡ ∼Q

Accepted Answer

To prove R ≡ ∼Q using an ordinary proof

Question 5

Given the following premises:
1)S ⊃ (∼∼T • ∼∼C)
2)(S • Q) ∨ C
3)∼C

Accepted Answer

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 
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

Question 6

Given the following premises:
1)E ⊃ (B • J)
2)(J • B) ⊃ ∼L
3)L

Accepted Answer

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 
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

Question 7

Given the following premises:
1)Q ⊃ (H • ∼F)
2)∼(Q • ∼M)
3)∼G ⊃ (Q • ∼M)

Accepted Answer

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 
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

Question 8

Given the following premises:
1)R ⊃ (∼B ⊃ F)
2)∼U ⊃ B
3)∼B

Accepted Answer

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 
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

Question 9

Given the following premises:
1)R ⊃ (E • D)
2)R • ∼G
3)∼E ⊃ G

Accepted Answer

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 
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

Question 10

Use indirect proof:
1.(P ∨ F) ⊃ (A ∨ D)
2.A ⊃ (M • ∼P)
3.D ⊃ (C • ∼P)
/ ∼P

Accepted Answer

To prove ∼P using an indirect proof (als

Question 11

Given the following premises:
1)N ≡ R
2)(N • ∼R) ⊃ C
3)N

Accepted Answer

A) (N ⊃ R) ∨ (R ⊃ N) 1, Equiv 
B) N • (∼R ⊃ 
C)  2, Assoc 
C) C ⊃ (N • ∼R) 2, Com 
D) N ⊃ (∼R ⊃ 
C)  2, Exp 
E) R 1, 3, MP 
A) (N ⊃ R) ∨ (R ⊃ N) 1, Equiv 
B) N • (∼R ⊃ 
C)  2, Assoc 
C) C ⊃ (N • ∼R) 2, Com 
D) N ⊃ (∼R ⊃ 
C)  2, Exp 
E) R 1, 3, MP

Question 12

Given the following premises:
1)T ⊃ (G ∨ G)
2)∼P ⊃ T
3)F ⊃ (B ⊃ ∼P)

Accepted Answer

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 
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

Question 13

Given the following premises:
1)(∼H • ∼J) ⊃ K
2)∼(∼H • ∼J)
3)(∼H • N) ∨ (∼H • ∼J)

Accepted Answer

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 
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

Question 14

Given the following premises:
1)S ∨ (∼Q ∨ ∼C)
2)(∼Q ∨ ∼C) ⊃ M
3)T ⊃ (Q • C)

Accepted Answer

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 
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

Question 15

Use indirect proof:
1.S ⊃ (R • ∼T)
2.(S • R) ⊃ (T ∨ E)
3.(Q ∨ ∼T) ⊃ ∼E
/ ∼S

Accepted Answer

To prove ∼S using indirect proof (also k

Question 16

Given the following premises:
1)E
2)R ⊃ ∼E
3)N ⊃ (∼C ⊃ R)

Accepted Answer

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 
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

Given the following premises: 1)∼R ∨ ∼R 2)R ∨ (∼J • ∼H) 3)∼R ⊃ (H • B)

Given the following premises: 1)Q ⊃ (∼N ∨ ∼N) 2)∼N ⊃ ∼∼P 3)P ⊃ ∼G

Given the following premises: 1)R • ∼S 2)R ⊃ ∼(S • ∼F) 3)∼S ⊃ (F • N)

Use an ordinary proof (not conditional or indirect proof): 1.A ⊃ (Q ∨ R) 2.(R • Q) ⊃ B 3.A • ∼B / R ≡ ∼Q

Given the following premises: 1)S ⊃ (∼∼T • ∼∼C) 2)(S • Q) ∨ C 3)∼C

Given the following premises: 1)E ⊃ (B • J) 2)(J • B) ⊃ ∼L 3)L

Given the following premises: 1)Q ⊃ (H • ∼F) 2)∼(Q • ∼M) 3)∼G ⊃ (Q • ∼M)

Given the following premises: 1)R ⊃ (∼B ⊃ F) 2)∼U ⊃ B 3)∼B

Given the following premises: 1)R ⊃ (E • D) 2)R • ∼G 3)∼E ⊃ G

Use indirect proof: 1.(P ∨ F) ⊃ (A ∨ D) 2.A ⊃ (M • ∼P) 3.D ⊃ (C • ∼P) / ∼P

Given the following premises: 1)N ≡ R 2)(N • ∼R) ⊃ C 3)N

Given the following premises: 1)T ⊃ (G ∨ G) 2)∼P ⊃ T 3)F ⊃ (B ⊃ ∼P)

Given the following premises: 1)(∼H • ∼J) ⊃ K 2)∼(∼H • ∼J) 3)(∼H • N) ∨ (∼H • ∼J)

Given the following premises: 1)S ∨ (∼Q ∨ ∼C) 2)(∼Q ∨ ∼C) ⊃ M 3)T ⊃ (Q • C)

Use indirect proof: 1.S ⊃ (R • ∼T) 2.(S • R) ⊃ (T ∨ E) 3.(Q ∨ ∼T) ⊃ ∼E / ∼S

Given the following premises: 1)E 2)R ⊃ ∼E 3)N ⊃ (∼C ⊃ R)

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Exam 7: Natural Deduction in Propositional Logic