Deck 7: Natural Deduction in Propositional Logic

Question 1

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

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

Answer 1

F ⊃ (∼B ∨ ∼P) 3, Impl

Question 2

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

Answer 2

Answer not provided

Question 3

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

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

Answer 3

S ⊃ W 3, Simp

Question 4

Given the following premises:
1)B
2)∼R ⊃ K
3)B ⊃ (K ⊃ E)

A)(B ⊃ K) ⊃ E 3, Assoc
B)∼R ⊃ E 2, 3, HS
C)R ∨ K 2, Impl
D)K ⊃ E 1, 3, MP
E)B • N 1, Add

Question 5

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

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 6

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

A)∼H ⊃ M 1, Impl
B)∼E 1, 2, MT
C)H 1, Simp
D)M ⊃ O 3, Simp
E)D ∨ O 1, 3, CD

Question 7

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

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 8

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

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 9

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

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 10

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

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

Question 11

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

Question 12

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

A)∼(C • L) 2, 3, MT
B)(C ⊃ ∼L) ∨ ∼N 1, Assoc
C)(C ⊃ ∼N) • (L ⊃ ∼N) 2, Dist
D)C ⊃ ∼N 2, Simp
E)C ⊃ ∼(L • N) 1, DM

Question 13

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

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 14

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

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 15

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

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 16

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

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 17

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

A)D ⊃ ∼A 1, Taut
B)D ⊃ A 1, DN
C)D ⊃ (R • M) 1, 2, HS
D)∼∼A 2, 3, MT
E)∼(R • M) 3, DM

Question 18

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

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 19

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

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

Question 20

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

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 21

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

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 22

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

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 23

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

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 24

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

Question 25

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

Question 26

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

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 27

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

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 28

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

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 29

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

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 30

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

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 31

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

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 32

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

Question 33

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

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 34

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

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 35

Use an ordinary proof (not conditional or indirect proof):
1.F ⊃ (J ∨ ∼F)
2.J ⊃ (L ∨ ∼J)
/ F ⊃ L

Question 36

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

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 37

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

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 38

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

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 39

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

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 40

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

A)∼(F • ∼M) 1, 3, MT
B)∼L 3, Simp
C)∼P 2, 3, MT
D)∼L ∨ G 3, DM
E)L ⊃ P 2, Trans

Question 41

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

Question 42

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

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 43

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

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 44

Use conditional proof:
1.N ⊃ (F • A)
2.B ⊃ (R • F)
/ (N ∨ B) ⊃ (A ∨ R)

Question 45

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

A)L ⊃ ∼(F ∨ L) 1, 2, HS
B)M ∨ J 1, 3, CD
C)L ⊃ M 1, Simp
D)∼M 2, 3, MT
E)M ⊃ (∼F ∨ ∼L) 2, DM

Question 46

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

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 47

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

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 48

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

Question 49

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

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 50

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

A)S ⊃ B 1, HS
B)∼F ∨ B 1, 2, CD
C)S 2, 3, DS
D)B 1, 3, MP
E)∼S 1, 3, MT

Question 51

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

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 52

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

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 53

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

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 54

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

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 55

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

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 56

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

Question 57

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

Question 58

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

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 59

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

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 60

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

Question 61

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

A)∼H 2, 3, DS
B)∼P • ∼(P ∨ ∼H) 2, 3, Conj
C)∼P • H 3, DM
D)E 1, 2, MT
E)∼P ⊃ E 1, Trans

Question 62

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

Question 63

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

Question 64

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

Question 65

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

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 66

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

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 67

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

Question 68

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

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 69

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

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 70

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

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 71

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

Question 72

Use natural deduction to prove the following logical truth:
[F • (D ⊃ ∼F)] ⊃ (D ⊃ A)

Question 73

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

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 74

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

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 75

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

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 76

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

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