Question 1

Given the following premises:
1)A
2)(A ⊃ ∼T) ⊃ ∼G
3)Q ⊃ (A ⊃ ∼T)

Accepted Answer

A) Q ⊃ (T ⊃ ∼
A)  3, Trans 
B) (Q ⊃ 
A)  ⊃ ∼T 3, Assoc 
C) A ⊃ (∼T • ∼G) 2, Exp 
D) ∼T 1, 3, MP 
E) Q ⊃ ∼G 2, 3, HS 
A) Q ⊃ (T ⊃ ∼
A)  3, Trans 
B) (Q ⊃ 
A)  ⊃ ∼T 3, Assoc 
C) A ⊃ (∼T • ∼G) 2, Exp 
D) ∼T 1, 3, MP 
E) Q ⊃ ∼G 2, 3, HS E

Question 2

Given the following premises:
1)∼W
2)C ∨ W
3)R ⊃ ∼(C ∨ W)

Accepted Answer

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 
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 A

Question 3

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

Accepted Answer

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 
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 E

Question 4

Given the following premises:
1)N ∨ C
2)(N ∨ C) ⊃ (F ⊃ C)
3)∼C

Accepted Answer

A) F ⊃ C 1, 2, MP 
B) N 1, 3, DS 
C) ∼F 2, 3, MT 
D) ∼N 1, 3, MT 
E) ∼C • R 3, Add 
A) F ⊃ C 1, 2, MP 
B) N 1, 3, DS 
C) ∼F 2, 3, MT 
D) ∼N 1, 3, MT 
E) ∼C • R 3, Add

Question 5

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

Question 6

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 7

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

Accepted Answer

A) (H • Q) ∨ (H • ∼S) 3, Dist 
B) ∼Q ∨ S 1, DM 
C) F 1, 2, MT 
D) H 1, 3, DS 
E) ∼∼F 1, 2, MT 
A) (H • Q) ∨ (H • ∼S) 3, Dist 
B) ∼Q ∨ S 1, DM 
C) F 1, 2, MT 
D) H 1, 3, DS 
E) ∼∼F 1, 2, MT

Question 8

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

Accepted Answer

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 
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

Question 9

Given the following premises:
1)∼M ⊃ S
2)∼M
3)(M ∨ H) ∨ ∼S

Accepted Answer

A) H 2, 3, DS 
B) M ∨ H 3, Simp 
C) M ∨ (H ∨ ∼S) 3, Assoc 
D) ∼S 1, 2, MP 
E) M ∨ S 1, Impl 
A) H 2, 3, DS 
B) M ∨ H 3, Simp 
C) M ∨ (H ∨ ∼S) 3, Assoc 
D) ∼S 1, 2, MP 
E) M ∨ S 1, Impl

Question 10

Given the following premises:
1)(G ⊃ A) ∨ T
2)G
3)∼T

Accepted Answer

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 
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

Question 11

Use an ordinary proof (not conditional or indirect proof):
1.E ⊃ (S ⊃ T)
2.(∼L • M) ⊃ (S • E)
3. ∼(T ∨ L)/ ∼M

Accepted Answer

To solve this problem, we will use an or

Question 12

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 13

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 14

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

Accepted Answer

A) ∼K ∨ H 3, DM 
B) K • ∼T 1, 3, DS 
C) K • (∼T ∨ ∼H) 1, Dist 
D) M 2, 3, MT 
E) (∼M • K) ⊃ ∼H 2, Exp 
A) ∼K ∨ H 3, DM 
B) K • ∼T 1, 3, DS 
C) K • (∼T ∨ ∼H) 1, Dist 
D) M 2, 3, MT 
E) (∼M • K) ⊃ ∼H 2, Exp

Question 15

Given the following premises:
1)F ⊃ J
2)A ⊃ (F • J)
3)A • (Q ∨ N)

Accepted Answer

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 
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

Question 16

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 17

Use conditional proof:
1.S ⊃ (B ⊃ T)
2.N ⊃ (T ⊃ ∼B)/ (S • N) ⊃ ∼B

Accepted Answer

To prove the argument using conditional

Question 18

Use natural deduction to prove the following logical truth:
[(P ∨ Q) ⊃ (R • T)] ⊃ (P ⊃ R)

Accepted Answer

To prove the logical truth of `[(P ∨ Q)

Question 19

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

Accepted Answer

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 
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

Question 20

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

Given the following premises: 1)A 2)(A ⊃ ∼T) ⊃ ∼G 3)Q ⊃ (A ⊃ ∼T)

Given the following premises: 1)∼W 2)C ∨ W 3)R ⊃ ∼(C ∨ W)

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

Given the following premises: 1)N ∨ C 2)(N ∨ C) ⊃ (F ⊃ C) 3)∼C

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

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

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

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

Given the following premises: 1)∼M ⊃ S 2)∼M 3)(M ∨ H) ∨ ∼S

Given the following premises: 1)(G ⊃ A) ∨ T 2)G 3)∼T

Use an ordinary proof (not conditional or indirect proof): 1.E ⊃ (S ⊃ T) 2.(∼L • M) ⊃ (S • E) 3. ∼(T ∨ L)/ ∼M

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

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

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

Given the following premises: 1)F ⊃ J 2)A ⊃ (F • J) 3)A • (Q ∨ N)

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

Use conditional proof: 1.S ⊃ (B ⊃ T) 2.N ⊃ (T ⊃ ∼B)/ (S • N) ⊃ ∼B

Use natural deduction to prove the following logical truth: [(P ∨ Q) ⊃ (R • T)] ⊃ (P ⊃ R)

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

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

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Exam 7: Natural Deduction in Propositional Logic