CHAPTER FIVE
Exercises 5.1.1E
1.a. Derive: A & B
1
2
A
A⊃Β
Assumption
Assumption
3
4
B
A&B
1, 2 ⊃E
1, 3 &I
c. Derive: A ⊃ (∼ C & ∼ B)
1
2
3
4
5
6
7
A ⊃ (∼ B & ∼ C)
Α
∼B&∼C
∼B
∼C
∼C&∼B
A ⊃ (∼ C & ∼ B)
Assumption
A / ⊃Ι
1, 2 ⊃E
3 &E
3 &E
4, 5 &I
1–6 ⊃Ι
e. Derive: ∼ A ⊃ [Β & (D & C)]
1
2
3
∼A⊃B
B⊃D
∼A⊃C
4
∼A
5
6
7
8
9
10
B
D
C
D&C
B & (D & C)
∼ A ⊃ [Β & (D & C)]
Assumption
Assumption
Assumption
A / ⊃Ι
1, 4 ⊃E
2, 5 ⊃E
3, 6 ⊃E
6, 7 &I
5, 8 &I
4–9 ⊃I
g. Derive: [(K ∨ L) ⊃ I ] & [(K ∨ L) ⊃ ∼ J]
1
(K ∨ L) ⊃ (I & ∼ J)
Assumption
2
K∨L
A / ⊃I
3
4
5
6
I&∼J
I
(K ∨ L) ⊃ I
K∨L
1, 2 ⊃E
3 &E
2–4 ⊃I
A / ⊃I
I&∼J
∼J
(K ∨ L) ⊃ ∼ J
[(K ∨ L) ⊃ I ] & [(K ∨ L) ⊃ ∼ J]
1, 6 ⊃E
7 &E
6–8 ⊃E
5, 9 &I
7
8
9
10
88 SOLUTIONS TO SELECTED EXERCISES ON PP. 168–169
i. Derive: A ⊃ (B ⊃ C)
1 (A & B) ⊃ C
2
Assumption
A / ⊃Ι
A
3
B
A / ⊃Ι
4
5
6
7
A&B
C
B⊃C
A ⊃ (B ⊃ C)
2, 3 & I
1, 4 ⊃E
3–5 ⊃I
2–6 ⊃I
k. Derive: (A & B) ⊃ (C & D)
1
2
3
4
5
6
7
8
9
10
(B & A) ⊃ (D & C)
A&B
B
A
B&A
D&C
C
D
C&D
(A & B) ⊃ (C & D)
Assumption
A / ⊃I
2 &E
2 &E
3, 4 &I
1, 5 ⊃E
6 &E
6 &E
7, 8 &I
2–9 ⊃I
m. Derive: (A & B) ⊃ E
1
2
3
A⊃C
B⊃D
(C & D) ⊃ E
4
A&B
5
6
7
8
9
10
11
A
B
C
D
C&D
E
(A & B) ⊃ E
Assumption
Assumption
Assumption
A / ⊃I
4 &E
4 &E
1, 5 ⊃E
2, 6 ⊃E
7, 8 &I
3, 9 ⊃E
4–10 ⊃1
Exercises 5.1.2E
1.a. Derive: ∼ G
1
(G ⊃ I) & ∼ I
Assumption
2
G
A /∼ I
3
4
5
6
G⊃I
I
∼I
∼G
1 &E
2, 3 ⊃E
1 &E
2–5 ∼ I
SOLUTIONS TO SELECTED EXERCISES ON PP. 168–169 and P. 175
89
c. Derive: ∼ ∼ B
1
2
∼B⊃A
∼B⊃∼A
3
∼B
4
5
6
A
∼A
∼∼B
Assumption
Assumption
A /∼ I
1, 3 ⊃E
2, 3 ⊃E
3–5 ∼ I
e. Derive: A
1
(∼ A ⊃ ∼ B)& (∼ B ⊃ B)
Assumption
2
∼A
A /∼ E
3
4
5
6
7
∼A⊃∼B
∼B
∼B⊃B
B
1 &E
2, 3 ⊃E
1 &E
4, 5 ⊃E
2–6 ∼ E
A
Exercises 5.1.3E
1.a. Derive: B ∨ (K ∨ G)
1
K
Assumption
2
3
K∨G
B ∨ (K ∨ G)
I ∨I
2 ∨I
c. Derive: D ∨ E
1
E∨D
2
E
A / ∨E
3
D∨E
2 ∨I
4
D
A / ∨E
5
6
D∨E
D∨E
Assumption
4∨I
1, 2–3, 4–5 ∨E
e. Derive: F
1
2
∼E∨F
∼E⊃F
Assumption
Assumption
3
∼E
A / ∨E
4
F
2, 3 ⊃ E
5
F
A / ∨E
F
5R
1, 3–4, 5–6 ∨E
6
7
F
90 SOLUTIONS TO SELECTED EXERCISES ON P. 175 and PP. 177–178
Exercises 5.1.4E
1.a. Derive: L
1
2
K (∼ E & L)
K
Assumption
Assumption
3
4
∼E&L
L
1, 2 E
3 &E
c. Derive: S & ∼ A
1
2
(S ∼ I) & N
(N ∼ I) & ∼ A
Assumption
Assumption
3
4
5
6
7
8
9
∼A
N∼I
N
∼I
S∼I
S
S&∼A
2 &E
2 &E
1 &E
4, 5 E
1 &E
6, 7 E
3, 8 &I
e. Derive: E O
1
2
(E ⊃ T) & (T ⊃ O)
O⊃E
Assumption
Assumption
3
E
A / I
4
5
6
7
E⊃T
T
T⊃O
O
1 &E
3, 4 ⊃E
1 &E
5, 6 ⊃E
8
O
A / I
9
10
E
EO
2, 8 ⊃E
3–7, 8–9 I
Exercises 5.3E
1. Derivability
a. Derive: A ⊃ (A & B)
1
G
A⊃B
Assumption
A ⊃ (A & B)
2–— ⊃I
SOLUTIONS TO SELECTED EXERCISES ON PP. 179–180 and PP. 223–228
91
Derive: A ⊃ (A & B)
1
2
A⊃B
A
Assumption
A / ⊃I
3
4
5
B
A&B
A ⊃ (A & B)
1, 2 ⊃E
2, 3 &I
2–5 ⊃I
c. Derive: L K
1
(K ⊃ L) & (L ⊃ K)
2
L
G
A / I
K
K
G
G
Assumption
L
LK
A / I
2–—, — –— I
Derive: L K
1
(K ⊃ L) & (L ⊃ K)
Assumption
2
L
A / I
3
4
L⊃K
K
1 &E
2, 3 ⊃E
5
K
A / I
6
7
8
K⊃L
L
LK
1 &E
5, 6 ⊃ E
2–4, 5–7 I
e. Derive: C
1
B&∼B
2
∼C
G
C
Assumption
A /∼ E
2–— ∼ E
92 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
Derive: C
1
B&∼B
2
∼C
A /∼ E
3
4
5
B
∼B
1 &E
1 &E
2–4 ∼ E
Assumption
C
g. Derive: D ⊃ B
1
2
A⊃C
(∼ A ∨ C) ⊃ (D ⊃ B)
Assumption
Assumption
G
G
∼A∨C
D⊃B
2, — ⊃E
Derive: D ⊃ B
1
2
3
4
5
6
7
8
9
10
11
12
A⊃C
(∼ A ∨ C) ⊃ (D ⊃ B)
∼ (∼ A ∨ C)
A
Assumption
Assumption
A /∼ E
A /∼ I
C
∼A∨C
∼ (∼ A ∨ C)
∼A
∼A∨C
∼ (∼ A ∨ C)
∼A∨C
D⊃B
1, 4 ⊃E
5 ∨I
3R
4–7 ∼I
8 ∨I
3R
3, 10 ∼ E
2, 11 ⊃E
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
93
i. Derive: B
1
2
A⊃B
∼ (B & ∼ C) ⊃ A
∼B
3
G
Assumption
Assumption
A /∼ E
3-— ∼ E
B
Derive: B
1
2
A⊃B
∼ (B & ∼ C) ⊃ A
∼B
3
B&∼C
4
5
6
7
8
9
10
11
B
∼B
∼ (B & ∼ C)
A
B
∼B
B
Assumption
Assumption
A /∼ E
A /∼ I
4 &E
3R
4–6 ∼ I
2, 7 ⊃E
1, 8 ⊃E
3R
3–10 ∼ E
k. Derive: B ∨ ∼ C
1
2
A ∨ (B & C)
C⊃∼A
3
A
G
B∨∼C
B&C
G
G
B∨∼C
B∨∼C
Assumption
Assumption
A / ∨E
A / ∨E
1, 3-—, —-— ∨E
94 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
Derive: B ∨ ∼ C
1
2
3
A ∨ (B & C)
C⊃∼A
Assumption
Assumption
A / ∨E
A
4
C
A /∼ I
5
6
7
8
∼A
A
∼C
B∨∼C
2, 4 ⊃E
3R
4–6 ∼ 1
7 ∨I
9
B&C
A / ∨E
10
11
12
B
B∨∼C
B∨∼C
9 &E
10 ∨I
1, 3–8, 9–11 ∨E
m. Derive: D ⊃ (F ⊃ C)
1
2
3
G
G
(A ∨ B) ⊃ C
(D ∨ E) ⊃ [(F ∨ G) ⊃ A]
A / ⊃I
D
F⊃C
D ⊃ (F ⊃ C)
3-— ⊃I
Derive: D ⊃ (F ⊃ C)
1
2
3
4
5
6
7
8
9
10
11
12
(A ∨ B) ⊃ C
(D ∨ E) ⊃ [(F ∨ G) ⊃ A]
A / ⊃I
D
A / ⊃I
F
D∨E
(F ∨ G) ⊃ A
F∨G
A
A∨B
C
F⊃C
D ⊃ (F ⊃ C)
3∨I
2, 5 ⊃E
4 ∨I
6, 7 ⊃E
8 ∨I
1, 9 ⊃E
4–10 ⊃I
3–11 ⊃I
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
95
o. Derive: B ⊃ F
1
2
3
A ⊃ ∼ (B ∨ C)
(C ∨ D) ⊃ A
∼ F ⊃ (D & ∼ E)
Β
4
G
G
Assumption
Assumption
Assumption
A / ⊃I
F
B⊃F
4-— ⊃I
Derive: B ⊃ F
1
2
3
A ⊃ ∼ (B ∨ C)
(C ∨ D) ⊃ A
∼ F ⊃ (D & ∼ E)
4
6
7
8
9
10
11
12
13
A / ⊃I
B
5
Assumption
Assumption
Assumption
∼F
A / ∼E
D&∼E
D
C∨D
A
∼ (B ∨ C)
B∨C
3, 5 ⊃ E
6 &E
7 ∨I
2, 8 ⊃ E
1, 9 ⊃E
4 ∨I
5–11 ∼ I
4–12 ⊃I
F
B⊃F
q. Derive: H
1
2
3
F ⊃ (G ∨ H)
∼ (∼ F ∨ H)
∼G
4
~Η
G
G
G
∼F∨H
~ (∼ F ∨ H)
H
Assumption
Assumption
Assumption
A /∼ E
2R
4-— ∼E
96 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
q. Derive: H
1
2
3
F ⊃ (G ∨ H)
∼ (∼ F ∨ H)
∼G
Assumption
Assumption
Assumption
4
∼H
A /∼ E
5
F
A/∼I
6
7
G∨H
G
1, 5 ⊃E
A / ∨E
8
∼H
A / ∼E
9
10
12
G
∼G
H
7R
3R
8–10 ∼ E
13
H
A / ∨E
14
15
16
17
18
19
20
21
∼H
H
4R
13 R
6, 7–12, 13–15 ∨E
4R
5–17 ∼E
18 ∨I
2R
4–20 ∼ E
H
∼H
∼F
∼F∨H
∼ (∼ F ∨ H)
H
2. Validity
a. Derive: A ⊃ C
1
2
A⊃∼B
∼B⊃C
3
A
A / ⊃I
4
5
6
∼B
C
A⊃C
1, 3 ⊃E
2, 4 ⊃E
3–5 ⊃I
Assumption
Assumption
c. Derive: ∼ B
1
2
AB
∼A
3
B
A/∼1
4
5
6
A
∼A
1, 3 E
2R
3–5 ∼ 1
∼B
Assumption
Assumption
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
97
e. Derive: A ⊃ [B ⊃ (C ⊃ D)]
1
2
D
Assumption
A / ⊃I
A
3
A / ⊃I
B
4
C
A / ⊃I
5
6
7
8
D
C⊃D
B ⊃ (C ⊃ D)
A ⊃ [B ⊃ (C ⊃ D)]
1R
4–5 ⊃I
3–6 ⊃I
2–7 ⊃I
g. Derive: A ⊃ (D ⊃ C)
1
2
3
A ⊃ (B ⊃ C)
D⊃B
Assumption
Assumption
A / ⊃I
A
A / ⊃I
4
D
5
6
7
8
9
B⊃C
B
C
D⊃C
A ⊃ (D ⊃ C)
1, 3 ⊃E
2, 4 ⊃E
5, 6 ⊃ E
4–7 ⊃I
3–8 ⊃I
i. Derive: A ⊃ C
1
2
∼A∨B
B⊃C
3
A
4
Assumption
Assumption
A / ⊃I
∼A
A / ∨E
5
∼C
A/∼E
6
7
8
A
∼A
C
3R
4R
5–7 ∼ E
9
B
A / ∨E
10
11
12
C
C
A⊃C
2, 9 ⊃E
1, 4–8, 9–10 ∨E
3–11 ⊃ I
98 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
k. Derive: B
1
2
3
A ⊃ (C ⊃ B)
∼C⊃∼A
A
Assumption
Assumption
Assumption
4
∼B
A/∼E
5
6
C⊃B
C
1, 3 ⊃E
A/∼I
7
8
9
10
11
12
B
∼B
∼C
∼A
A
5, 6 ⊃ E
4R
6–8 ∼ I
2, 9 ⊃E
3R
4–11 ∼ E
B
*m. Derive: F & G
1
2
FG
F∨G
3
F
A / ∨E
4
F
3R
5
G
A / ∨E
6
7
8
9
Assumption
Assumption
F
F
G
F&G
1,
2,
1,
7,
5 E
3–4, 5–6 ∨E
7 E
8 &I
3. Theorems
a. Derive: A ⊃ (A ∨ B)
1
2
3
A / ⊃I
A
A∨B
A ⊃ (A ∨ B)
1 ∨I
1–2 ⊃I
c. Derive: A ⊃ [B ⊃ (A & B)]
1
2
3
4
5
A / ⊃I
A
A / ⊃Ι
B
A&B
B ⊃ (A & B)
A ⊃ [B ⊃ (A & B)]
1, 2 &I
2–3 ⊃I
1–4 ⊃I
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
99
e. Derive: (A B) ⊃ (A ⊃ B)
1
AB
A / ⊃I
2
A
A / ⊃I
3
4
5
B
A⊃B
(A B) ⊃ (A ⊃ B)
1, 2 E
2–3 ⊃I
1–4 ⊃I
g. Derive: (A ⊃ B) ⊃ [(C ⊃ A) ⊃ (C ⊃ B)]
1
A⊃B
A / ⊃I
2
C⊃A
A / ⊃I
3
C
A / ⊃I
4
5
6
7
8
A
B
C⊃B
(C ⊃ A) ⊃ (C ⊃ B)
(A ⊃ B) ⊃ [(C ⊃ A) ⊃ (C ⊃ B)]
2, 3 ⊃E
1, 4 ⊃E
3–5 ⊃I
2–6 ⊃I
1–7 ⊃I
i. Derive: [(A ⊃ B) & ∼ B] ⊃ ∼ A
1
(A ⊃ B) & ∼ B
2
3
4
5
6
7
A / ⊃I
A / ⊃I
A
A⊃B
B
∼B
∼A
[(A ⊃ B) & ∼ B] ⊃ ∼ A
1 &E
2, 3 ⊃I
1 &E
2–5 ∼ I
1–6 ⊃I
k. Derive: A ⊃ [B ⊃ (A ⊃ B)]
1
2
A / ⊃I
A
A / ⊃I
B
3
A
A / ⊃I
4
5
6
7
B
A⊃B
B ⊃ (A ⊃ B)
A ⊃ [B ⊃ (A ⊃ B)]
2R
3–4 ⊃I
2–5 ⊃I
1–6 ⊃I
100 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
m. Derive: (A ⊃ B) ⊃ [∼ B ⊃ ∼ (A & D)]
1
A⊃B
A / ⊃I
2
∼B
A / ⊃I
3
4
5
6
7
8
9
A&D
A
B
∼B
∼ (A & D)
∼ B ⊃ ∼ (A & D)
(A ⊃ B) ⊃ [∼ B ⊃ ∼ (A & D)]
A/∼I
3 &E
1, 4 ⊃E
2R
3–6 ∼ I
2–7 ⊃I
1–8 ⊃I
4. Equivalence
a. Derive: A & ∼ A
1
2
3
4
5
B&∼B
Assumption
∼ (A & ∼ A)
B
∼B
A&∼A
A/∼E
1 &E
1 &E
2–4 ∼ E
Derive: B & ∼ B
1
2
3
4
5
A&∼A
Assumption
∼ (B & ∼ B)
A
∼A
B&∼B
A/∼E
1 &E
1 &E
2–4 ∼ E
c. Derive: (A ∨ B) ⊃ A
1
2
B⊃A
Assumption
A∨B
A / ⊃I
3
A
A / ∨E
4
A
3R
5
B
A / ∨E
6
7
8
A
A
(A ∨ B) ⊃ A
1, 5 ⊃E
2, 3–4, 5–6 ∨E
2–7 ⊃I
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
101
Derive: B ⊃ A
1
2
3
4
5
(A ∨ B) ⊃ A
Assumption
A / ⊃I
B
A∨B
A
B⊃A
2 ∨I
1, 3 ⊃E
2–4 ⊃I
e. Derive: ∼ (A B)
1
2
3
(A & ∼ B) ∨ (B & ∼ A)
A&∼B
AB
Assumption
A / ∨E
A/∼I
4
5
6
7
A
B
∼B
∼ (A B)
2 &E
3, 4 E
2 &E
2–6 ∼ I
8
B&∼A
A / ∨E
9
10
11
12
13
14
AB
B
A
∼A
∼ (A B)
∼ (A B)
A/∼I
8 &E
9, 10 E
8&E
9–12 ∼ I
1, 2–6, 7–13 ∨E
Derive: (A & ∼ B) ∨ (B & ∼ A)
1
2
3
∼ (A B)
∼ [(A & ∼ B) ∨ (B & ∼ A)]
Assumption
A/∼I
A / I
A
4
∼B
A/∼E
5
6
7
8
A&∼B
(A & ∼ B) ∨ (B & ∼ A)
∼ [(A & ∼ B) ∨ (B & ∼ A)]
B
3, 4 &I
5 ∨I
2R
4–7 ∼ I
9
B
A / ⊃I
10
∼A
A/∼E
11
12
13
14
15
14
15
B&∼A
(A & ∼ B) ∨ (B & ∼ A)
∼ [(A & ∼ B) ∨ (B & ∼ A)]
9, 10 &I
11 ∨ I
2R
10–13 ∼ E
3–8, 9–14 I
1R
2–14 ∼ E
A
AB
∼ (A B)
(A & ∼ B) ∨ (B & ∼ A)
102 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
5. Inconsistency
a. Derive: A ⊃ A, ∼ (A ⊃ A)
1
2
3
4
5
∼ (A ⊃ A)
Assumption
A/⊃I
A
A
A⊃A
∼ (A ⊃ A)
2R
2–3 ⊃I
1R
c. Derive: A, ∼ A
1
2
3
AB
B⊃∼A
A
Assumption
Assumption
Assumption
4
5
6
A
B
∼A
3R
1, 4 E
2, 5 ⊃E
e. Derive: A, ∼ A
1
2
A⊃∼A
∼A⊃A
3
A
A/∼I
4
5
6
7
∼A
A
∼A
A
1, 3 ⊃E
3R
A/∼I
2, 6 ⊃E
Assumption
Assumption
g. Derive: A ∨ B, ∼ (A ∨ B)
1
2
3
4
5
6
7
8
9
10
11
∼ (A ∨ B)
C⊃A
∼C⊃A
Assumption
Assumption
Assumption
A/∼I
C
A
A∨B
∼ (A ∨ B)
∼C
B
A∨B
∼ (A ∨ B)
2, 4 ⊃E
5 ∨I
1R
4–7 ∼ I
3,8 ⊃E
9 ∨I
1R
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
103
i. Derive: F ∨ G, ∼ (F ∨ G)
1
2
3
∼ (F ∨ G) (A ⊃ A)
H⊃F
∼H⊃F
4
5
6
7
8
9
10
11
12
13
14
Assumption
Assumption
Assumption
A / ⊃I
A
A
A⊃A
∼ (F ∨ G)
H
4R
4–5 ⊃ I
1, 6 E
A/∼I
F
F∨G
∼ (F ∨ G)
∼H
F
F∨G
2, 8 ⊃E
9 ∨I
7R
8–11 ∼ I
3, 12 ⊃E
13 ∨I
6. Derivability
a. Derive: A B
1
2
A⊃B
∼A⊃∼B
Assumption
Assumption
3
A
A / I
4
B
1, 3 ⊃E
5
B
A/I
6
∼A
A/∼E
7
8
9
10
∼B
B
2, 6 ⊃E
5R
6–8 ∼ E
3–4, 5–9 I
A
AB
c. Derive: A
1
2
A (∼ B ∨ C)
B⊃C
∼A
3
4
5
6
7
8
9
10
11
12
13
B
C
∼B∨C
A
∼A
∼B
∼B∨C
A
∼A
A
104 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
Assumption
Assumption
A/∼E
A/∼I
2, 4 ⊃E
5 ∨I
1, 6 E
3R
4–8 ∼ I
9∨I
1, 10 E
3R
3–12 ∼ E
e. Derive: B ∨ D
1
2
3
B ∨ (C ∨ D)
C⊃A
A⊃∼C
Assumption
Assumption
Assumption
4
B
A / ∨E
5
B∨D
4 ∨I
6
C∨D
A / ∨E
7
C
A / ∨E
∼ (B ∨ D)
8
A/∼E
9
10
11
12
A
∼C
C
B∨D
2, 7 ⊃E
3, 9 ⊃E
7R
8–11 ∼ E
13
D
A / ∨E
14
15
16
B∨D
B∨D
B∨D
13 ∨I
6, 7–12, 13–14 ∨E
1, 4–5, 6–15 ∨E
g. Derive: (A ∨ B) ⊃ ∼ C
1
2
3
4
A ⊃ (D & B)
(∼ D B) & (C ⊃ A)
A∨B
A / ⊃I
A / ∨E
A
5
Assumption
Assumption
A/∼I
C
6
7
8
9
10
11
D&B
∼DB
B
∼D
D
∼C
1, 4 ⊃E
2 &E
6 &E
7, 8 E
6 &E
5–10 ∼ I
12
B
A / ∨E
13
14
15
16
17
18
19
20
21
22
23
A/∼I
C
C⊃A
A
D&B
D
∼DB
B
∼D
∼C
∼C
(A ∨ B) ⊃ ∼ C
2 &E
13, 14 ⊃E
1, 15 ⊃E
16 &E
2 &E
16 &E
18, 19 E
13–20 ∼ I
3, 4–11, 12–21 ∨E
2–22 ⊃I
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
105
7. Validity
a. Derive: ∼ (C ∼ A)
1
∼ (C ∨ A)
Assumption
2
C∼A
A/∼I
3
∼A
A/∼E
C
C∨A
∼ (C ∨ A)
2, 3 E
4 ∨I
1R
3–6 ∼ E
7 ∨I
1R
2–9 ∼ I
4
5
6
7
8
9
10
A
C∨A
∼ (C ∨ A)
∼ (C ∼ A)
c. Derive: A B
1
2
∼A&∼B
Assumption
A / I
A
3
∼B
A/∼E
4
5
6
∼A
A
B
1 &E
2R
3–5 ∼ E
7
B
A / I
8
∼A
A/∼E
9
10
11
12
∼B
B
1 &E
7R
8–10 ∼ E
2–6, 7–11 I
A
AB
106 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
e. Derive: ∼ H
1
2
3
H ∼ (I & ∼ J)
∼I∼H
J⊃∼I
Assumption
Assumption
Assumption
4
H
A/∼I
5
6
∼ (I & ∼ J)
∼I
1, 4 E
A/∼E
7
8
9
10
∼H
H
2, 6 E
4R
6–8 ∼ E
A/∼I
11
12
13
14
15
I
J
∼I
I
3, 10 ⊃E
9R
10–13 ∼ I
9, 13 & I
4–14 ∼ I
∼J
I&∼J
∼H
g. Derive: H ∨ ∼ I
1
2
3
(F ∨ G) ∨ (H ∨ ∼ I)
F⊃H
I⊃∼G
Assumption
Assumption
Assumption
4
F∨G
A / ∨E
5
F
A/∨E
6
7
H
H∨∼I
2, 5 ⊃E
6 ∨I
8
G
A / ∨E
9
I
A/∼I
∼G
G
10
11
12
13
14
∼I
H∨∼I
H∨∼I
3, 9 ⊃ E
8R
9–11 ∼ I
12 ∨I
4, 5–7, 8–13 ∨E
15
H∨∼I
A / ∨E
16
17
H∨∼I
H∨∼I
15 R
1, 4–14, 15–16 ∨E
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
107
i. Derive: F ∨ (I & ∼ G)
1
2
∼ (F ∨ ∼ G) ∼ (H ∨ I)
F∨I
Assumption
Assumption
3
F
A / ∨E
5
6
F ∨ (I & ∼ G)
I
3∨I
A / ∨E
7
8
9
10
11
∼ (F ∨ ∼ G)
∼ (H ∨ I)
H∨I
F∨∼G
F
A /∼ E
1, 7 E
6 ∨I
7–9 ∼ E
A / ∨E
12
F ∨ (I & ∼ G)
11 ∨I
13
∼G
A / ∨E
14
15
16
17
I&∼G
F ∨ (I & ∼ G)
F ∨ (I & ∼ G)
F ∨ (I & ∼ G)
6, 13 &I
15 ∨I
10, 11–12, 13–15 ∨E
2, 3–5, 6–16 ∨E
k. Derive: (∼ A ∼ C) ⊃ (∼ A D)
1
2
3
(∼ A ∼ C) (B ∼ D)
∼A⊃∼B
C⊃∼D
Assumption
Assumption
Assumption
4
∼A∼C
A / ⊃I
5
∼A
A / I
6
∼D
A/∼E
B∼D
B
∼B
7
8
9
10
D
1, 4 ⊃E
6, 7 E
2, 5 ⊃E
6–9 ∼ E
11
D
A / I
12
13
14
15
16
17
20
C
∼D
D
∼C
∼A
∼AD
(∼ A ∼ C) ⊃ (∼ A D)
108 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
A/∼I
3, 12 ⊃E
11 R
12–14 ∼ I
4, 15 E
5–10, 11–16 I
4–17 ⊃I
m. Derive: ∼ E
1
2
3
∼ (A ⊃ B) & (C & ∼ D)
(B ∨ ∼ A) ∨ [(C & E) ⊃ D]
Assumption
Assumption
A/∼I
E
4
B∨∼A
A / ∨E
5
B
A / ∨E
6
A
A / ⊃I
7
8
B
A⊃B
5R
6–7 ⊃ I
9
∼A
A / ∨E
10
A / ⊃I
A
11
∼B
A/∼E
12
13
14
15
16
A
∼A
B
A⊃B
A⊃B
10 R
9R
11–13 ∼ E
10–14 ⊃I
4, 5–8, 9–15 ∨E
17
(C & E) ⊃ D
A / ∨E
18
∼ (A ⊃ B)
A/∼E
19
20
21
22
22
23
24
25
26
C&∼D
∼D
C
C&E
D
A⊃B
A⊃B
∼ (A ⊃ B)
∼E
1 &E
19 &E
19 &E
3, 21 &I
17, 22 ⊃E
18–22 ∼ E
2, 4–16, 17–23 ∨E
1 &E
3–25 ∼ I
9. Theorems
a. Derive: ∼ (A ⊃ B) ⊃ ∼ (A B)
1
∼ (A ⊃ B)
A / ⊃I
2
AB
A/∼I
3
A
A / ⊃I
4
5
6
7
8
B
A⊃B
∼ (A ⊃ B)
∼ (A B)
∼ (A ⊃ B) ⊃ ∼ (A B)
2, 3 E
3–4 ⊃I
1R
2–6 ∼ I
1–7 ⊃I
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
109
c. Derive: (A ⊃ B) ∨ (B ⊃ A)
1
2
∼ [(A ⊃ B) ∨ (B ⊃ A)]
4
5
6
7
8
9
A/∼I
B
3
A/∼E
A
B
A⊃B
(A ⊃ B) ∨ (B ⊃ A)
∼ [(A ⊃ B) ∨ (B ⊃ A)]
∼B
B
A / ⊃I
2R
3–4 ⊃I
5 ∨I
1R
2–7 ∨ I
A / ⊃I
10
∼A
A/∼E
11
12
13
14
15
16
17
B
∼B
9R
8R
10–12 ∼ E
9–13 ⊃I
14 ∨I
1R
1–16 ∼ E
A
B⊃A
(A ⊃ B) ∨ (B ⊃ A)
∼ [(A ⊃ B) ∨ (B ⊃ A)]
(A ⊃ B) ∨ (B ⊃ A)
e. Derive: [(A ∨ B) ⊃ C] [(A ⊃ C) & (B ⊃ C)]
1
2
(A ∨ B) ⊃ C
A / I
A / ⊃I
A
A∨B
C
A⊃C
B
2 ∨I
1, 3 ⊃E
2–4 ⊃I
A / ⊃I
7
8
9
10
A∨B
C
B⊃C
(A ⊃ C) & (B ⊃ C)
6 ∨I
1, 7 ⊃E
6–8 ⊃I
5, 9 &I
11
(A ⊃ C) & (B ⊃ C)
A / I
3
4
5
6
12
A∨B
A / ⊃I
13
A
A / ∨E
14
15
A⊃C
C
11 &E
13, 14 ⊃E
16
B
A / ∨E
17
18
19
20
21
B⊃C
C
11 &E
16, 17 ⊃E
12, 13–15, 16–18 ∨E
12–19 ⊃I
1–10, 11–20 I
C
(A ∨ B) ⊃ C
[(A ∨ B) ⊃ C] [(A ⊃ C) & (B ⊃ C)]
110 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
g. Derive: ∼ (A B) (A ∼ B)
1
2
3
∼ (A B)
A / I
A / I
A
A/∼I
B
4
A
A / I
5
B
3R
6
B
A / I
7
8
9
10
A
AB
∼ (A B)
∼B
2R
4–5, 6–7 I
1R
3–9 ∼ I
11
∼B
A / I
12
∼A
13
A/∼E
A / I
A
14
∼B
A/∼E
15
16
17
A
∼A
B
13 R
12 R
14–16 ∼ E
18
B
A / I
19
∼A
A/∼E
20
21
22
23
24
25
26
B
∼B
A
A∼B
18 R
11 R
19–21 ∼ E
13–17, 18–22
1R
12–24 ∼ E
2–10, 11–25 I
27
A∼B
A / I
A
AB
∼ (A B)
28
AB
A/∼I
29
B
A/∼I
30
31
32
33
34
A
∼B
B
∼B
∼B
28, 29 E
27, 30 E
29 R
29–32 ∼ I
A/∼E
35
36
37
38
39
40
A
B
∼B
27, 34 E
28, 35 E
34 R
34–37 ∼ E
28–38 ∼ I
B
∼ (A B)
∼ (A & B)
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
111
10. Equivalence
a. Derive: ∼ ∼ A
1
A
Assumption
2
∼A
A/∼I
3
4
5
A
∼A
∼∼A
1R
2R
2–4 ∼ I
Derive: A
1
∼∼A
Assumption
2
∼A
A/∼E
3
4
5
∼A
∼∼A
2R
1R
2–4 ∼ E
A
c. Derive: A ∨ A
1
A
Assumption
2
A∨A
A / ∨I
Derive: A ∨ A
1
A∨A
2
3
4
Assumption
A
A / ∨E
A
2R
1, 2–3, 2–3 ∨E
A
e. Derive: B ∨ A
1
A∨B
Assumption
2
A
A / ∨E
3
B∨A
2 ∨I
4
Β
A / ∨E
5
6
B∨A
B∨A
4 ∨I
1, 2–3, 4–5 ∨E
112 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
Derive: A ∨ B
1
B∨A
Assumption
2
B
A / ∨E
3
A∨B
2 ∨I
4
Α
A / ∨E
5
6
A∨B
A∨B
4 ∨I
1, 2–3, 4–5 ∨E
g. Derive: (A ∨ B) ∨ C
1
A ∨ (B ∨ C)
Assumption
2
A
A / ∨E
3
4
A∨B
(A ∨ B) ∨ C
2 ∨I
3 ∨I
5
B∨C
A / ∨E
6
B
A / ∨E
7
9
A∨B
(A ∨ B) ∨ C
6 ∨I
7 ∨I
C
A / ∨I
10
11
12
13
(A ∨ B) ∨ C
(A ∨ B) ∨ C
(A ∨ B) ∨ C
10 ∨I
5, 6–9, 10–11 ∨I
1, 2–4, 5–12 ∨E
Derive: A ∨ (B ∨ C)
1
(A ∨ B) ∨ C
2
A∨B
Assumption
A / ∨E
3
A
A / ∨E
4
A ∨ (B ∨ C)
3 ∨I
5
B
A / ∨E
6
7
8
9
10
11
12
B∨C
A ∨ (B ∨ C)
A ∨ (B ∨ C)
C
B∨C
A ∨ (B ∨ C)
A ∨ (B ∨ C)
5 ∨I
6 ∨I
2, 3–4, 5–7 ∨E
A / ∨E
9 ∨I
10 ∨I
1, 2–8, 9–11 ∨E
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
113
i. Derive: ∼ B ⊃ ∼ A
1
A⊃B
2
∼B
3
4
5
6
7
Assumption
A / ⊃I
A/∼E
A
B
∼B
∼A
∼B⊃∼A
1, 3 ⊃E
2R
3–5 ∼ I
2–6 ⊃I
Derive: A ⊃ B
1
2
∼B⊃∼A
Α
Assumption
A / ⊃I
3
∼B
A/∼E
4
5
6
7
A
∼A
2R
1, 3 ⊃E
3–5 ∼ E
2–6 ⊃I
Β
A⊃B
k. Derive: A B
1
(A & B) ∨ (∼ A & ∼ B)
Assumption
2
A&B
A / ∨E
3
A
A / I
4
B
2 &E
5
B
A / I
6
7
A
AB
2 &E
3–4, 5–6 I
8
∼A&∼B
A / ∨E
9
A / I
A
10
∼B
A/∼E
11
12
13
A
∼A
B
9R
8 &E
10–12 ∼ E
14
B
A / I
15
∼A
A/∼E
16
17
18
19
20
B
∼B
14 R
8 &E
15–17 ∼ E
9–13, 14–18 I
1, 2–7, 8–19 ∨E
A
AB
AB
114 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
Derive: (A & B) ∨ (∼ A & ∼ B)
1
2
AB
∼ [(A & B) ∨ (∼ A & ∼ B)]
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Assumption
A/∼E
A/∼I
A
B
A&B
(A & B) ∨ (∼ A & ∼ B)
∼ [(A & B) ∨ (∼ A & ∼ B)]
∼A
B
A
∼A
∼B
∼A&∼B
(A & B) ∨ (∼ A & ∼ B)
∼ [(A & B) ∨ (∼ A & ∼ B)]
(A & B) ∨ (∼ A & ∼ B)
1, 3 E
3, 4 &I
5∨I
2R
3–7 ∼ I
A/∼I
1, 9 E
8R
9–11 ∼ I
8, 12 &I
13 ∨I
2R
2–15 ∼ E
m. Derive: (A ∨ B) & (A ∨ C)
1
A ∨ (B & C)
Assumption
2
A
A / ∨E
3
4
5
A∨B
A∨C
(A ∨ B) & (A ∨ C)
2 ∨I
2 ∨I
3, 4 &I
6
B&C
A / ∨E
7
8
9
10
11
12
B
A∨B
C
A∨C
(A ∨ B) & (A ∨ C)
(A ∨ B) & (A ∨ C)
6&E
7 ∨I
6 &E
9 ∨I
8, 10 &I
1, 2–5, 6–11 ∨E
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
115
Derive: A ∨ (B & C)
1
(A ∨ B) & (A ∨ C)
Assumption
2
3
A∨B
A
1 &E
A / ∨E
4
A ∨ (B & C)
3 ∨I
5
B
A / ∨E
6
7
A∨C
A
1 &E
A / ∨E
8
A ∨ (B & C)
7 ∨I
9
C
A / ∨E
10
11
12
13
B&C
A ∨ (B & C)
A ∨ (B & C)
A ∨ (B & C)
5, 9 &I
10 ∨I
6, 7–8, 9–11 ∨E
2, 3–4, 5–12 ∨E
o. Derive: ∼ A ∨ ∼ B
1
2
∼ (A & B)
∼ (∼ A ∨ ∼ B)
Assumption
A/∼E
3
∼A
A/∼E
4
5
6
7
∼A∨∼B
∼ (∼ A ∨ ∼ B)
3 ∨I
2R
3–5 ∼ E
A/∼E
8
9
10
11
12
13
A
∼B
∼A∨∼B
∼ (∼ A ∨ ∼ B)
B
A&B
∼ (A & B)
∼A∨∼B
7 ∨I
2R
7–9 ∼ E
6, 10 &I
1R
2–12 ∼ E
116 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
Derive: ∼ (A & B)
1
∼A∨∼B
Assumption
2
A&B
A/∼I
3
∼A
A / ∨E
4
∼A
3R
5
∼B
A / ∨E
6
7
8
9
10
11
12
A/∼I
A
B
∼B
∼A
∼A
A
∼ (A & B)
2 &E
5R
6–8 ∼ I
1, 3–4, 5–9 ∨E
2 &E
2–11 ∼ I
12. Inconsistency
a. Derive: B, ∼ B
1
2
(A ⊃ B) & (A ⊃ ∼ B)
(C ⊃ A) & (∼ C ⊃ A)
Assumption
Assumption
3
4
5
A⊃B
A⊃∼B
C
1 &E
1 &E
A/∼I
6
7
8
9
10
11
12
13
14
C⊃A
A
B
∼B
∼C
∼C⊃A
A
B
∼B
2 &E
5, 6 ⊃E
3, 7 ⊃E
4, 7 ⊃E
5–9 ∼ I
2 &E
10, 11 ⊃E
3, 12 ⊃E
4, 12 ⊃E
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
117
c. Derive: A, ∼ A
1
2
C∨A
CA
3
A
A/∼I
4
5
6
7
8
C
∼A
A
2, 3 E
1, 4 E
3R
3–6 ∼ I
A/∼E
9
10
11
12
∼A
∼A
C
A
∼A
A
Assumption
Assumption
1, 8 E
2, 9 E
8R
8–11 ∼ E
e. Derive: A, ∼ A
1
2
∼ [(A ∨ B) ∨ C ]
A∼C
Assumption
Assumption
3
A
A/∼1
4
5
6
7
8
A∨B
(A ∨ B) ∨ C
∼ [(A ∨ B) ∨ C ]
3 ∨I
4 ∨I
1R
3–6 ∼ I
A/∼E
∼A
∼A
9
C
10
11
12
13
14
15
A
∼A
∼C
A
∼A
A
A/∼I
2, 9 E
8R
9–11 ∼ I
2, 12 E
8R
8–14 ∼ E
118 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
g. Derive: B, ∼ B
1
2
3
A & (B ∨ C)
(∼ C ∨ H) & (H ⊃ ∼ H)
∼B
Assumption
Assumption
Assumption
4
B∨C
1 &E
5
B
A / ∨E
6
B
5R
7
C
A / ∨E
8
9
∼C∨H
∼C
2 &E
A / ∨E
10
∼B
A/∼E
11
12
13
C
∼C
B
7R
9R
10–12 ∼ E
14
H
A / ∨E
15
∼B
A/∼E
16
17
18
19
20
21
H⊃∼H
∼H
2 &E
14, 16 ⊃E
15, 17 ∼ E
8, 9–13, 14–18 ∨E
4, 5–6, 7–19 ∨E
3R
B
B
B
∼B
13. Validity
a. Derive: M
1
2
3
S&F
F⊃B
(B & ∼ M) ⊃ ∼ S
4
5
6
7
8
9
10
M
Assumption
Assumption
Assumption
∼M
A/∼E
F
B
B&∼M
∼S
S
1 &E
2, 5 ⊃E
6, 4 &I
3, 7 ⊃E
1 &E
4–9 ∼ E
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
119
c. Derive: ∼ J
1
2
3
(C ⊃ ∼ R) & (R ⊃ L)
C (C ∨ L)
J⊃R
4
5
6
7
8
9
10
11
12
∼J
Assumption
Assumption
Assumption
J
A/∼I
R
R⊃L
L
C∨L
C
C⊃∼R
∼R
3, 4 ⊃E
1 &E
5, 6 ⊃E
7 ∨I
2, 8 E
1 &E
9, 10 ⊃E
4–11 ∼ I
e. Derive: ∼ M
1
2
3
∼ (R ∨ W)
(R M) ∨ [(M ∨ G) ⊃ (W M)]
Assumption
Assumption
A/∼I
M
4
RM
A / ∨E
5
6
R
R∨W
3, 4 E
5 ∨I
7
(M ∨ G) ⊃ (W M)
A / ∨E
8
9
10
11
12
13
14
M∨G
WM
W
R∨W
R∨W
∼ (R ∨ W)
∼M
120 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
3 ∨I
7, 8 ⊃E
3, 9 E
10 ∨I
2, 4–6, 7–11 ∨E
1R
3–13 ∼ I
g. Derive: H ⊃ J
1
2
3
4
(H & T) ⊃ J
(M ⊃ D) & (∼ D ⊃ M)
∼ T (∼ D & M)
∼J
6
7
8
9
10
11
12
13
14
15
16
17
A / ⊃I
H
5
A/∼E
A/∼I
T
H&T
J
∼J
∼T
∼D&M
M⊃D
M
D
∼D
J
H⊃J
Assumption
Assumption
Assumption
4, 6 &I
1, 7 ⊃E
5R
6–9 ∼ I
3, 10 E
2 &E
11 &E
12, 13 ⊃E
11 &E
5–15 ∼ E
4–16 ⊃I
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
121
i. Derive: L ⊃ T
1
2
L ⊃ (C ∨ T)
(∼ L ∨ B) & (∼ B ∨ ∼ C)
Assumption
Assumption
3
L
A / ⊃I
4
5
C∨T
C
1, 3 ⊃E
A / ∨E
∼B∨∼C
∼B
6
7
∼L∨B
∼L
8
9
2 &E
A / ∨E
2 &E
A / ∨E
10
∼T
A/∼E
11
12
13
L
∼L
T
3R
9R
10–12 ∼ E
14
B
A / ∨E
15
∼T
A/∼E
16
17
18
19
B
∼B
T
14 R
7R
15–17 ∼ E
8, 9–13, 14–18 ∨E
20
∼C
A / ∨E
T
21
∼T
A/∼E
22
23
24
25
∼C
C
T
20 R
5R
21–23 ∼ E
6, 7–19, 20–24 ∨E
26
T
A / ∨E
T
26 R
4, 5–25, 26–27 ∨E
3–28 ⊃ I
27
28
29
T
T
L⊃T
122 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
14. Inconsistency
a. 1
2
3
4
5
6
7
8
c. 1
2
3
4
5
6
7
8
9
10
11
e. 1
2
3
(M ⊃ B) & (B ⊃ P)
M&∼P
Assumption
Assumption
M
M⊃B
B
B⊃P
P
∼P
2 &E
1 &E
3, 4 ⊃E
1 &E
5, 6 ⊃E
2 &E
B⊃I
(∼ B & ∼ I) ⊃ C
∼C&∼I
Assumption
Assumption
Assumption
A/∼I
B
I
∼I
∼B
∼I
∼B&∼I
C
∼C
1, 4 ⊃E
3 &E
4–6 ∼ I
3 &E
7, 8 &I
2, 9 ⊃E
3 &E
M ∨ (F ⊃ T)
N∼T
(F & N) & ∼ M
Assumption
Assumption
Assumption
4
M
A / ∨E
5
M
4R
6
F⊃T
A / ∨E
7
8
9
10
11
12
13
14
15
M
M
∼M
∼M
A/∼E
F&N
F
T
N
∼T
3 &E
8 &E
6, 9 ⊃E
8 &E
2, 11 E
7–12 ∼ E
1, 4–5, 6–13 ∨E
3 &E
15.a. We do not want this rule as a rule of SD because it is not truthpreserving. The truth of P ∨ Q does not entail the truth of P.
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
123
c. We can show that Reiteration is dispensable by explaining how do
derive P whenever P occurs on an earlier accessible line, without using Reiteration. Assume that P occurs on an accessibleline i and that we want to derive
P on a later line. We can do this as follows:
i
n
n1
P
P&P
P
i, i &I
n &E
e. Assume that P is a theorem is SD. Now considere any argument
has ∼ P as one of its premises:
Q1
Q2
Qn
∼P
R
We can derive R from the set consisting of the premises as follows:
1
2
Q1
Q2
Assumption
Assumption
n
Qn
Assumption
n1
∼P
Assumption
n2
∼R
n3
∼P
n4
nk
nk1
A/∼E
n1R
P
R
n2nk∼E
Here lines n 4 through n k consists of the derivation of P from no primary
assumptions. We know there is such a derivation because we know P is a theorem of SD.
e. If P is a theorem is SD then any argument of SL that has ∼ P among
its premises is vald in SD. We can construct a derivation of the conclusion,
call it Q by taking the premises of the argument as our primary assumptions.
124 SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228
∼ P will be one of these assumptions. Next assume ∼ Q, derive both P and
∼ P, and obtain Q by Negation Elimination. We can obtain ∼ P by Reiteration
since it is one of the primary assumptions of the derivation. We can obtain P
because it is a theorem of SD and therefore can be derived from the empty
set. If it can be derived from the empty set it can also be derived from the set
consisting of the premises of the argument, by inserting the derivation of P
from the empty set within the scope of the assumption ∼ Q.
16. We here make use of a result established in Sections 6.3 and 6.4,
|– P in SD if and only if |= P
a. Assume that a given argument is valid in SD. Then we know that its
conclusion is derivable in SD from the set consisting of its premises. By the
above result it follows that the conclusion of the argument is truth-functionally
entailed by the set consisting of the premises of the argument. Therefore there
is no truth-value assignment on which the members of the set, which are just
the premises of the argument, are true and the conclusion of the argument
false. So the argument is truth-functionally valid. Conversely, assume that the
given argument is truth-functionally valid. So there is no truth-value assignment
on which the premises of the argument are true and the conclusion false. From
this it follows that the set consisting of the premises of the argument truthfunctionally entails the conclusion of the argument. And by the above result
it next follows that the conclusion of the argument is derivable from the set
consisting of the premises of the argument, and from this it follows that the
argument is valid in SD.
d. Assume that sentences P and Q of SL are equivalent in SD. Then
each can be derived from the unit set of the other. By the above result it follows that the unit set of each truth-functionally entails the other. So there is
no truth-value assignment on which P is true and Q false, and no truth-value
assignment on which Q is true and P false. So P and Q are truth-functionally
equivalent.
Exercises 5.4E
Derive: ∼ D
1
2
3
D⊃E
E ⊃ (Z & W)
∼Z∨∼W
Assumption
Assumption
Assumption
4
5
6
∼ (Z & W)
∼E
∼D
3 DeM
2, 4 MT
1, 5 MT
SOLUTIONS TO SELECTED EXERCISES ON PP. 223–228 and PP. 236–239
125
c. Derive: K
1
2
3
(W ⊃ S) & ∼ M
(∼ W ⊃ H) ∨ M
(∼ S ⊃ H) ⊃ K
Assumption
Assumption
Assumption
4
5
6
7
8
9
W⊃S
∼S⊃∼W
∼M
∼W⊃H
∼S⊃H
K
1 &E
4 Trans
1 &E
2, 6 DS
5, 7 HS
3, 8 ⊃E
e. Derive: C
1
2
(M ∨ B) ∨ (C ∨ G)
∼ B & (∼ G & ∼ M)
Assumption
Assumption
3
4
5
6
7
8
9
10
11
12
∼B
(B ∨ M) ∨ (C ∨ G)
B ∨ [M ∨ (C ∨ G)]
M ∨ (C ∨ G)
∼G&∼M
∼G
(M ∨ C) ∨ G
M∨C
∼M
C
2 &E
1 Com
4 Assoc
3, 5 DS
2 &E
7 &E
6 Assoc
8, 9 DS
7 &E
10, 11 DS
2. Validity
a. Derive: Y Z
1
2
3
∼Y⊃∼Z
∼Z⊃∼X
∼X⊃∼Y
Assumption
Assumption
Assumption
4
Y
A / E
5
6
7
∼Z⊃∼Y
Y⊃Z
Z
2, 3 HS
5 Trans
4, 6 ⊃E
8
Z
A / I
9
10
11
Z⊃Y
Y
YZ
126 SOLUTIONS TO SELECTED EXERCISES ON PP. 236–239
1 Trans
8, 9 ⊃E
4–7, 8–10 I
c. Derive: I ⊃ ∼ D
1
2
3
4
5
6
7
8
9
10
11
12
13
(F & G) ∨ (H & ∼ I)
I ⊃ ∼ (F & D)
Assumption
Assumption
A / ⊃I
I
∼ (F & D)
∼F∨∼D
∼∼I
∼H∨∼∼I
∼ (H & ∼ I)
F&G
F
∼∼F
∼D
I⊃∼D
2, 3 ⊃E
4 DeM
3 DN
6 ∨I
7 DeM
1, 8 DS
9 &E
10 DN
5, 11 DS
3–12 ⊃I
e. Derive: I ∨ H
1
2
3
F ⊃ (G ⊃ H)
∼ I ⊃ (F ∨ H)
F⊃G
Assumption
Assumption
Assumption
4
∼I
A / ⊃I
5
6
F∨H
∼H
2, 4 ⊃E
A/∼E
7
8
9
10
11
12
13
14
F
G
G⊃H
∼G
H
∼I⊃H
∼∼I∨H
I∨H
5, 6 DS
3, 7 ⊃E
1, 7 ⊃E
6, 9 MT
6–10 ∼ E
4–11 ⊃I
12 Impl
13 DN
SOLUTIONS TO SELECTED EXERCISES ON PP. 236–239
127
g. Derive: X Y
1
2
3
[(X & Z) & Y] ∨ (∼ X ⊃ ∼ Y)
X⊃Z
Z⊃Y
Assumption
Assumption
Assumption
4
X
A / I
5
6
Z
Y
2, 4 ⊃E
3, 5 ⊃E
7
Y
A / I
(X & Z) & Y
A / ∨E
9
10
X&Z
X
8 &E
9 &E
11
∼X⊃∼Y
A / ∨E
12
13
14
15
Y⊃X
X
11 Trans
7, 12 ⊃E
1, 8–10, 11–13 ∨E
4–6, 7–14 I
8
X
XY
3. Theorems
a. Derive: A ∨ ∼ A
1
∼ (A ∨ ∼ A)
A/∼E
2
3
4
5
∼A&∼∼A
∼A
∼∼A
A∨∼A
1 DeM
2 &E
2 &E
1–4 ∼ E
c. Derive: A ∨ [(∼ A ∨ B) & (∼ A ∨ C)]
1
2
3
4
5
6
∼A
∼ A ∨ (B & C)
(∼ A ∨ B) & (∼ A ∨ C)
∼ A ⊃ [(∼ A ∨ B) & (∼ A ∨ C)]
∼ ∼ A ∨ [(∼ A ∨ B) & (∼ A ∨ C)]
A ∨ [(∼ A ∨ B) & (∼ A ∨ C)]
128 SOLUTIONS TO SELECTED EXERCISES ON PP. 236–239
A / ⊃I
1 ∨I
2 Dist
1–3 ⊃I
4 Impl
5 DN
e. Derive: [A ⊃ (B & C)] [(∼ B ∨ ∼ C) ⊃ ∼ A]
1
A ⊃ (B & C)
A / I
2
3
∼ (B & C) ⊃ ∼ A
(∼ B ∨ ∼ C) ⊃ ∼ A
1 Trans
2 DeM
4
(∼ B ∨ ∼ C) ⊃ ∼ A
A / I
5
6
7
∼ (B & C) ⊃ ∼ A
A ⊃ (B & C)
[A ⊃ (B & C)] [(∼ B ∨ ∼ C) ⊃ ∼ A]
4 DeM
5 Trans
1–3, 4–6 I
g. Derive: [A ⊃ (B C)] (A ⊃ [(∼ B ∨ C) & (∼ C ∨ B)])
1
A ⊃ (B C)
A / I
2
3
4
A ⊃ [(B ⊃ C) & (C ⊃ B)]
A ⊃ [(∼ B ∨ C) & (C ⊃ B)]
A ⊃ [(∼ B ∨ C) & (∼ C ∨ B)]
1 Equiv
2 Impl
3 Impl
5
A ⊃ [(∼ B ∨ C) & (∼ C ∨ B)]
A / I
6
7
8
9
A ⊃ [(B ⊃ C) & (∼ C ∨ B)]
A ⊃ [(B ⊃ C) & (C ⊃ B)]
A ⊃ (B C)
[A ⊃ (B C)] (A ⊃ [(∼ B ∨ C) & (∼ C ∨ B)])
5 Impl
6 Impl
7 Equiv
1–4, 5–8 I
i. Derive: [∼ A ⊃ (∼ B ⊃ C)] ⊃ [(A ∨ B) ∨ (∼ ∼ B ∨ C)]
1
2
3
4
5
6
7
8
9
∼ A ⊃ (∼ B ⊃ C)
∼ ∼ A ∨ (∼ B ⊃ C)
∼ ∼ A ∨ (∼ ∼ B ∨ C)
A ∨ (∼ ∼ B ∨ C)
A ∨ [(∼ ∼ B ∨ ∼ ∼ B) ∨ C]
A ∨ [∼ ∼ B ∨ (∼ ∼ B ∨ C)]
(A ∨ ∼ ∼ B) ∨ (∼ ∼ B ∨ C)
(A ∨ B) ∨ (∼ ∼ B ∨ C)
[∼ A ⊃ (∼ B ⊃ C)] ⊃ [(A ∨ B) ∨ (∼ ∼ B ∨ C)]
A / ⊃I
1 Impl
2 Impl
3 DN
4 Idem
5 Assoc
6 Assoc
7 DN
1–8 ⊃I
4. Equivalence
a. Derive: ∼ (∼ A & ∼ B)
1
A∨B
Assumption
2
3
4
∼∼A∨B
∼∼A∨∼∼B
∼ (∼ A & ∼ B)
1 DN
2 DN
3 DeM
SOLUTIONS TO SELECTED EXERCISES ON PP. 236–239
129
Derive: A ∨ B
1
∼ (∼ A & ∼ B)
Assumption
2
3
4
∼∼A∨∼∼B
A∨∼∼B
A∨B
1 DeM
2 DN
3 DN
c. Derive: ∼ (A ⊃ C) ⊃ ∼ B
1
(A & B) ⊃ C
Assumption
2
3
4
(B & A) ⊃ C
B ⊃ (A ⊃ C)
∼ (A ⊃ C) ⊃ ∼ B
1 Com
2 Exp
3 Trans
Derive: (A & B) ⊃ C
1
∼ (A ⊃ C) ⊃ ∼ B
Assumption
2
3
4
B ⊃ (A ⊃ C)
(B & A) ⊃ C
(A & B) ⊃ C
1 Trans
2 Exp
3 Com
e. Derive: A ∨ (∼ B ∼ C)
1
A ∨ (B C)
2
3
4
5
6
A
A
A
A
A
∨
∨
∨
∨
∨
[(B ⊃ C) & (C ⊃ B)]
[(∼ C ⊃ ∼ B) & (C ⊃ B)]
[(∼ C ⊃ ∼ B) & (∼ B ⊃ ∼ C)]
[(∼ B ⊃ ∼ C) & (∼ C ⊃ ∼ B)]
(∼ B ∼ C)
Assumption
1
2
3
4
5
Equiv
Trans
Trans
Com
Equiv
Derive: A ∨ (B C)
1
A ∨ (∼ B ∼ C)
2
3
4
5
6
A
A
A
A
A
∨
∨
∨
∨
∨
[(∼ B ⊃ ∼ C) & (∼ C ⊃ ∼ B)]
[(C ⊃ B) & (∼ C ⊃ ∼ B)]
[(C ⊃ B) & (B ⊃ C)]
[(B ⊃ C) & (C ⊃ B)]
(B C)
Assumption
1
2
3
4
5
Equiv
Trans
Trans
Com
Equiv
5. Inconsistency
a. 1
2
3
4
5
6
7
[(E & F) ∨ ∼ ∼ G] ⊃ M
∼ [[(G ∨ E) & (F ∨ G)] ⊃ (M & M)]
Assumption
Assumption
∼
∼
∼
∼
∼
2
3
4
5
6
([(G ∨ E) & (F ∨ G)] ⊃ M)
([(G ∨ E) & (G ∨ F)] ⊃ M)
([G ∨ (E & F)] ⊃ M)
([(E & F) ∨ G] ⊃ M)
([(E & F) ∨ ∼ ∼ G] ⊃ M)
130 SOLUTIONS TO SELECTED EXERCISES ON PP. 236–239
Idem
Com
Dist
Com
DN
c. 1
2
3
4
M&L
[L & (M & ∼ S)] ⊃ K
∼K∨∼S
∼ (K ∼ S)
Assumption
Assumption
Assumption
Assumption
K⊃∼S
[(L & M) & ∼ S] ⊃ K
(L & M) ⊃ (∼ S ⊃ K)
L&M
∼S⊃K
(K ⊃ ∼ S) & (∼ S ⊃ K)
K∼S
3 Impl
2 Assoc
6 Exp
1 Com
7, 8 ⊃E
5, 9 &I
10 Equiv
e. 1
2
3
∼ [W & (Z ∨ Y)]
(Z ⊃ Y) ⊃ Z
(Y ⊃ Z) ⊃ W
Assumption
Assumption
Assumption
4
5
∼ W ∨ ∼ (Z ∨ Y)
∼Z
1 DeM
A/∼E
5
6
7
8
9
10
11
6
7
8
9
10
11
12
13
14
15
16
17
18
∼
∼
∼
∼
∼
Z
Z
∼
∼
∼
∼
∼
∼
(Z ⊃ Y)
(∼ Z ∨ Y)
∼Z&∼Y
∼Z
Z
∨Y
∼ (Z ∨ Z)
W
(Y ⊃ Z)
(∼ Y ∨ Z)
∼Y&∼Z
Z
2, 5 MT
6 Impl
7 DeM
8 &E
5R
5–10 ∼ E
11 ∨I
12 DN
4, 13 DS
3, 14 MT
15 Impl
16 DeM
17 &E
6. Validity
a. Derive: ∼ B
1
2
3
4
5
6
7
8
9
10
(R ⊃ C) ∨ (B ⊃ C)
∼ (E & A) ⊃ ∼ (R ⊃ C)
∼E&∼C
Assumption
Assumption
Assumption
∼E
∼E∨∼A
∼ (E & A)
∼ (R ⊃ C)
B⊃C
∼C
∼B
3 &E
4 ∨I
5 DeM
2, 6 ⊃E
1, 7 DS
3 &E
8, 9 MT
SOLUTIONS TO SELECTED EXERCISES ON PP. 236–239
131
c. Derive: ∼ W ⊃ ∼ A
1
2
A ⊃ [W ∨ ∼ (C ∨ R)]
∼R⊃C
Assumption
Assumption
3
∼W
A / ⊃I
4
A
A/∼I
5
6
7
8
9
10
11
W ∨ ∼ (C ∨ R)
∼ (C ∨ R)
∼∼R∨C
R∨C
C∨R
∼A
∼W⊃∼A
1, 4 ⊃E
3, 5 DS
2 Impl
7 DN
8 Com
4–9 ∼ I
3–10 ⊃I
e. Derive: J ⊃ ∼ (E ∨ ∼ M)
1
2
3
4
5
6
7
8
9
10
11
12
13
∼ (J & ∼ H)
∼H∨M
E⊃∼M
Assumption
Assumption
Assumption
A / ⊃I
J
∼J∨∼∼H
∼∼J
∼∼H
M
∼∼M
∼E
∼E&∼∼M
∼ (E ∨ ∼ M)
J ⊃ ∼ (E ∨ ∼ M)
1 DeM
4 DN
5, 6 DS
2, 7 DS
8 DN
3, 9 MT
10, 9 &I
11 DeM
4–12 ⊃I
g. Derive: ∼ A ⊃ [H ⊃ (F & B)]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
(H & ∼ S) ⊃ A
∼B⊃∼S
∼S∨C
C⊃F
∼A
H
H ⊃ (∼ S ⊃ A)
∼S⊃A
∼∼S
C
F
∼∼B
B
F&B
H ⊃ (F & B)
∼ A ⊃ [H ⊃ (F & B)]
132 SOLUTIONS TO SELECTED EXERCISES ON PP. 236–239
Assumption
Assumption
Assumption
Assumption
A / ⊃I
A / ⊃I
1 Exp
6, 7 ⊃E
5, 8 MT
3, 9 DS
4, 10 ⊃E
2, 9 MT
12 DN
11, 13 &I
6–14 ⊃I
5–15 ⊃I
7. Inconsistency
a. 1
2
3
4
5
6
7
8
9
10
B∨∼C
(L ⊃ ∼ G) ⊃ C
(G ∼ B) & (∼ L ⊃ ∼ B)
∼L
Assumption
Assumption
Assumption
Assumption
∼L∨∼G
L⊃∼G
C
∼L⊃∼B
∼B
∼C
4 ∨I
5 Impl
2, 6 ⊃E
3 &E
4, 8 ⊃E
1, 9 DS
8.a. The rules of replacement are two-way rules. If we can derive Q from
P by using only these rules, we can derive P from Q by using the rules in
reverse order.
c. Suppose that before a current line n of a derivation, an accessible
line i contains a sentence of the form P ⊃ Q. The sentence P ⊃ (P & Q) can
be derived by using the following routine:
i
n
n1
n2
n3
P⊃Q
P
Assumption
Q
P&Q
P ⊃ (P & Q)
i, n ⊃E
n, n 1 &E
n n 2 ⊃I
SOLUTIONS TO SELECTED EXERCISES ON PP. 236–239
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