Question 1

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 2

Given the following premises:
1)(E ⊃ K) ∨ W
2)∼W
3)W ∨ ∼(Q ⊃ E)

Accepted Answer

A) E ⊃ K 1, 2, DS 
B) Q ⊃ K 1, 3, HS 
C) ∼(Q ⊃ 
E)  2, 3, DS 
D) E ⊃ (K ∨ W) 1, Assoc 
E) W ∨ (∼Q ⊃ ∼
E)  3, DM 
A) E ⊃ K 1, 2, DS 
B) Q ⊃ K 1, 3, HS 
C) ∼(Q ⊃ 
E)  2, 3, DS 
D) E ⊃ (K ∨ W) 1, Assoc 
E) W ∨ (∼Q ⊃ ∼
E)  3, DM

Question 3

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

Question 4

Given the following premises:
1)(C • ∼F) ⊃ E
2)G ∨ (C • ∼F)
3)∼(C • ∼F)

Accepted Answer

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. 
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.

Question 5

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 
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

Question 6

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 7

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 8

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 
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 9

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 10

Use an ordinary proof (not conditional or indirect proof):
1.S ⊃ (K • F)
2.F ⊃ (G • H)
/ S ⊃ H

Accepted Answer

To prove that S ⊃ H using an ordinary pr

Question 11

Given the following premises:
1)∼P
2)L ⊃ (P ∨ M)
3)(P • M) ⊃ (∼R ∨ ∼R)

Accepted Answer

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 
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

Question 12

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

Accepted Answer

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 
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

Question 13

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

Accepted Answer

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

Question 14

Given the following premises:
1)Q ⊃ (H • L)
2)H ⊃ ∼Q
3)L ⊃ ∼Q

Accepted Answer

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 
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

Question 15

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 16

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

Question 17

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 18

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 19

Given the following premises:
1)(S • ∼J) ∨ (∼S • ∼∼J)
2)S ∨ ∼S
3)∼J ⊃ P

Accepted Answer

A) S 2, Taut 
B) ∼J ∨ ∼∼J 1, 2, CD 
C) S ≡ ∼J 1, Equiv 
D) J ∨ P 3, Impl 
E) ∼P ⊃ J 3, Trans 
A) S 2, Taut 
B) ∼J ∨ ∼∼J 1, 2, CD 
C) S ≡ ∼J 1, Equiv 
D) J ∨ P 3, Impl 
E) ∼P ⊃ J 3, Trans

Question 20

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 
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

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

Given the following premises: 1)(E ⊃ K) ∨ W 2)∼W 3)W ∨ ∼(Q ⊃ E)

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

Given the following premises: 1)(C • ∼F) ⊃ E 2)G ∨ (C • ∼F) 3)∼(C • ∼F)

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

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

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

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

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

Use an ordinary proof (not conditional or indirect proof): 1.S ⊃ (K • F) 2.F ⊃ (G • H) / S ⊃ H

Given the following premises: 1)∼P 2)L ⊃ (P ∨ M) 3)(P • M) ⊃ (∼R ∨ ∼R)

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

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

Given the following premises: 1)Q ⊃ (H • L) 2)H ⊃ ∼Q 3)L ⊃ ∼Q

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

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

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

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

Given the following premises: 1)(S • ∼J) ∨ (∼S • ∼∼J) 2)S ∨ ∼S 3)∼J ⊃ P

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

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Exam 7: Natural Deduction in Propositional Logic