Question 1

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

Question 2

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

Accepted Answer

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

Question 3

Use an ordinary proof (not conditional or indirect proof):
1.K ∨ (S • N)
2.∼(K • ∼Q)
3.∼(N • ∼Q)/ Q

Accepted Answer

To prove Q using an ordinary proof with ...

Question 4

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

Accepted Answer

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

Question 5

Use conditional proof:
1.G ⊃ (E ⊃ N)
2.H ⊃ (∼N ⊃ E)/ G ⊃ (H ⊃ N)

Accepted Answer

To prove the argument using conditional ...

Question 6

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

Question 7

Given the following premises:
1) P ⊃ L
2) ∼(J • O) 
3) (L ⊃ A)  ⊃ (J • O)

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

Use an ordinary proof (not conditional or indirect proof):
1.∼N ⊃ (∼R ⊃ C)
2.R ⊃ N
3.∼C/ N

Accepted Answer

To prove that N is true using an ordinar...

Question 10

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

Accepted Answer

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

Question 11

Given the following premises:
1) (S ⊃ R)  ⊃ (J ⊃ T) 
2) (P ⊃ R)  ⊃ (S ⊃ R) 
3) R ⊃ J

Accepted Answer

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

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

Question 13

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

Question 14

Given the following premises:
1) ∼U ⊃ (S • K) 
2) R ⊃ (∼U • ∼U) 
3) S ≡ ∼U

Accepted Answer

A)  (∼U • S)  ⊃ K 1, Exp
B)  R ⊃ U 2, DN
C)  R ⊃ ∼U 2, Taut
D)  R ⊃ (S • K)  1, 2, HS
E)  (S ⊃ U)  • (∼U ⊃ ∼S)  3, Equiv

Question 15

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

Accepted Answer

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

Question 16

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

Question 17

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

Accepted Answer

A)  Q ⊃ (∼∼A ∨ ∼T)  1, DN
B)  (A ∨ ∼T)  ⊃ Q 1, Com
C)  (Q ⊃ 
A)   ∨ ∼T 1, Assoc
D)  Q 1, 3, MP
E)  A 2, 3, DS

Question 18

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

Question 19

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

Accepted Answer

A)  A ∨ G 3, DN
B)  (G ⊃ 
A)   ⊃ ∼L 2, Assoc
C)  ∼L 1, 2, MP
D)  ∼G 1, 3, DS
E)  G ⊃ (∼∼L ⊃ ∼
A)   2, Trans

Question 20

Given the following premises:
1) D ⊃ H
2) ∼D
3) ˜(D • S)

Accepted Answer

A)  ∼H 1, 2, MT
B)  ∼D ∨ (D ⊃ H)  2, Add
C)  H ⊃ D 1, Com
D)  S 2, 3, DS
E)  ∼D • ∼S 3, DM

Exam 7: Natural Deduction in Propositional Logic

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

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Use conditional proof: 1.G ⊃ (E ⊃ N) 2.H ⊃ (∼N ⊃ E)/ G ⊃ (H ⊃ N)

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Use an ordinary proof (not conditional or indirect proof): 1.K ∨ (S • N) 2.∼(K • ∼Q) 3.∼(N • ∼Q)/ Q

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

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

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

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

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

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

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Given the following premises: 1) D ⊃ H 2) ∼D 3) ˜(D • S)

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Given the following premises: 1) ∼U ⊃ (S • K) 2) R ⊃ (∼U • ∼U) 3) S ≡ ∼U

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

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

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

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Use an ordinary proof (not conditional or indirect proof): 1.∼N ⊃ (∼R ⊃ C) 2.R ⊃ N 3.∼C/ N

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Given the following premises: 1) P ⊃ L 2) ∼(J • O) 3) (L ⊃ A) ⊃ (J • O)

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

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Given the following premises: 1) (S ⊃ R) ⊃ (J ⊃ T) 2) (P ⊃ R) ⊃ (S ⊃ R) 3) R ⊃ J

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

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

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