Question 1

Given the following premises:
1)(J • ∼N) ∨ T
2)∼(J • ∼N)
3)∼T

Accepted Answer

A) T 1, 2, DS 
B) ∼J ∨ N 2, DM 
C) J • ∼N 1, 3, DS 
D) J • (∼N ∨ T) 1, Assoc 
E) ∼J 2, Simp 
A) T 1, 2, DS 
B) ∼J ∨ N 2, DM 
C) J • ∼N 1, 3, DS 
D) J • (∼N ∨ T) 1, Assoc 
E) ∼J 2, Simp

Question 2

Given the following premises:
1)∼I ∨ ∼∼B
2)M ⊃ ∼I
3)I

Accepted Answer

A) M ⊃ ∼∼B 1, 2, HS 
B) ∼∼B 1, 3, DS 
C) ∼M 2, 3, MT 
D) ∼I ⊃ M 2, Com 
E) ∼(I • ∼
B)  1, DM 
A) M ⊃ ∼∼B 1, 2, HS 
B) ∼∼B 1, 3, DS 
C) ∼M 2, 3, MT 
D) ∼I ⊃ M 2, Com 
E) ∼(I • ∼
B)  1, DM

Question 3

Use an ordinary proof (not conditional or indirect proof):
1.K ⊃ L
2.∼K ∨ F
3.(L • F) ⊃ A
4.∼A/ ∼K

Accepted Answer

To prove ∼K using an ordinary proof with

Question 4

Use an ordinary proof (not conditional or indirect proof):
1.M ⊃ (R • E)
2.(E ∨ H) ⊃ G/ M ⊃ G

Accepted Answer

To prove that M ⊃ G using an ordinary pr

Question 5

Use an ordinary proof (not conditional or indirect proof):
1.G ⊃ (H ⊃ K)
2.(H ∨ ∼M) ⊃ ∼K
3.H/ ∼G

Accepted Answer

To prove ∼G using an ordinary proof with

Question 6

Use indirect proof:
1.(R ∨ S) ⊃ (H • ∼G)
2.(K ∨ R) ⊃ (G ∨ ∼H)/ ∼R

Accepted Answer

To prove ∼R (not R) using an indirect pr

Question 7

Given the following premises:
1)∼D ∨ ∼T
2)D ∨ (∼T • ∼R)
3)D

Accepted Answer

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 
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

Question 8

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

Accepted Answer

A) ∼K 1, 2, MT 
B) (U ∨ K) • ∼N 3, Assoc 
C) (K • N) ⊃ F 2, Exp 
D) (U ∨ K) • (U ∨ ∼N) 3, Dist 
E) ∼(N • F) 1, DM 
A) ∼K 1, 2, MT 
B) (U ∨ K) • ∼N 3, Assoc 
C) (K • N) ⊃ F 2, Exp 
D) (U ∨ K) • (U ∨ ∼N) 3, Dist 
E) ∼(N • F) 1, DM

Question 9

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 10

Given the following premises:
1)G • ˜A
2)K ⊃ (G • ˜A)
3)G ⊃ M

Accepted Answer

A) (K ⊃ G ) ⊃ ˜A 2, Exp 
B) K ⊃ (˜A • G) 2, Com 
C) (K ⊃ G) • ˜A 2, Assoc 
D) K 1, 2, MP 
E) M 1, 3, MP 
A) (K ⊃ G ) ⊃ ˜A 2, Exp 
B) K ⊃ (˜A • G) 2, Com 
C) (K ⊃ G) • ˜A 2, Assoc 
D) K 1, 2, MP 
E) M 1, 3, MP

Question 11

Given the following premises:
1)P • (∼H ∨ D)
2)∼(∼P • ∼H)
3)(P ⊃ ∼H) • (∼P ⊃ H)

Accepted Answer

A) P ≡ ∼H 3, Equiv 
B) ∼H ∨ D 1, Simp 
C) (P • ∼H) ∨ D 1, Assoc 
D) P • (H ⊃ 
D)  1, Impl 
E) P • H 2, DN 
A) P ≡ ∼H 3, Equiv 
B) ∼H ∨ D 1, Simp 
C) (P • ∼H) ∨ D 1, Assoc 
D) P • (H ⊃ 
D)  1, Impl 
E) P • H 2, DN

Question 12

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 13

Use natural deduction to prove the following logical truth:
(P ⊃ Q) ≡ [P ⊃ (Q ∨ ∼P)]

Accepted Answer

To prove the logical truth (P ⊃ Q) ≡ [P

Question 14

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 15

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

Accepted Answer

A) ∼C 3, Simp 
B) T ⊃ ∼N 2, 3, HS 
C) (∼C • T) ⊃ R 3, Assoc 
D) ∼R 1, 2, MT 
E) N ⊃ ∼R 2, Trans 
A) ∼C 3, Simp 
B) T ⊃ ∼N 2, 3, HS 
C) (∼C • T) ⊃ R 3, Assoc 
D) ∼R 1, 2, MT 
E) N ⊃ ∼R 2, Trans

Question 16

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

Accepted Answer

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 
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

Given the following premises: 1)(J • ∼N) ∨ T 2)∼(J • ∼N) 3)∼T

Given the following premises: 1)∼I ∨ ∼∼B 2)M ⊃ ∼I 3)I

Use an ordinary proof (not conditional or indirect proof): 1.K ⊃ L 2.∼K ∨ F 3.(L • F) ⊃ A 4.∼A/ ∼K

Use an ordinary proof (not conditional or indirect proof): 1.M ⊃ (R • E) 2.(E ∨ H) ⊃ G/ M ⊃ G

Use an ordinary proof (not conditional or indirect proof): 1.G ⊃ (H ⊃ K) 2.(H ∨ ∼M) ⊃ ∼K 3.H/ ∼G

Use indirect proof: 1.(R ∨ S) ⊃ (H • ∼G) 2.(K ∨ R) ⊃ (G ∨ ∼H)/ ∼R

Given the following premises: 1)∼D ∨ ∼T 2)D ∨ (∼T • ∼R) 3)D

Given the following premises: 1)∼N • ∼F 2)K ⊃ (N • F) 3)U ∨ (K • ∼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)G • ˜A 2)K ⊃ (G • ˜A) 3)G ⊃ M

Given the following premises: 1)P • (∼H ∨ D) 2)∼(∼P • ∼H) 3)(P ⊃ ∼H) • (∼P ⊃ H)

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

Use natural deduction to prove the following logical truth: (P ⊃ Q) ≡ [P ⊃ (Q ∨ ∼P)]

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

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

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

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Exam 7: Natural Deduction in Propositional Logic