yay derivations

derivations level 1

level 2

level 3

level 4

100

label the missing justifications

a. Derive: A & B

1 A Assumption

2 A ⊃ B Assumption

3 B

4 A & B

1,2 ⊃E

1, 3 &I

100

1.a. Derive: ∼ G

1 (G ⊃ I) & ∼ I Assumption

2 G A /∼ I

3 G ⊃ I 1 &E

4 I 2, 3 ⊃E

5 ∼ I 1 &E

6 ∼ G 2–5 ∼ I

100

1.a. Derive: B ∨ (K ∨ G)

1 K Assumption

1.a. Derive: B ∨ (K ∨ G)

1 K Assumption

2 K ∨ G I ∨I

3 B ∨ (K ∨ G) 2 ∨I

100

a. Derive: A ⊃ (A ∨ B)

1 A A / ⊃I

2 A ∨ B 1 ∨I

3 A ⊃ (A ∨ B) 1–2 ⊃I

200

label the missing justifications:

1 A ⊃ (∼ B & ∼ C) Assumption

2 Α

3 ∼ B & ∼ C

4 ∼ B

5 ∼ C

6 ∼ C & ∼ B

7 A ⊃ (∼ C & ∼ B)

A / ⊃Ι

1, 2 ⊃E

3 &E

4, 5 &I

1–6 ⊃Ι

200

c. Derive: ∼ ∼ B

1 ∼ B ⊃ A Assumption

2 ∼ B ⊃ ∼ A Assumption

c. Derive: ∼ ∼ B

1 ∼ B ⊃ A Assumption

2 ∼ B ⊃ ∼ A Assumption

3 ∼ B A /∼ I

4 A 1, 3 ⊃E

5 ∼ A 2, 3 ⊃E

6 ∼ ∼ B 3–5 ∼ I

200

e. Derive: F

1 ∼ E ∨ F Assumption

2 ∼ E ⊃ F Assumption

e. Derive: F

1 ∼ E ∨ F Assumption

2 ∼ E ⊃ F Assumption

3 ∼ E A / ∨E

4 F 2, 3 ⊃ E

5 F A / ∨E

6 F 5 R

7 F 1, 3–4, 5–6 ∨E

200

Derive: A ⊃ [B ⊃ (A & B)]

c. Derive: A ⊃ [B ⊃ (A & B)]

1 A A / ⊃I

2 B A / ⊃Ι

3 A & B 1, 2 &I

4 B ⊃ (A & B) 2–3 ⊃I

5 A ⊃ [B ⊃ (A & B)] 1–4 ⊃I

300

label the missing justifications

Derive: ∼ A ⊃ [Β & (D & C)]

1 ∼ A ⊃ B Assumption

2 B ⊃ D Assumption

3 ∼ A ⊃ C Assumption

4 ∼ A A / ⊃Ι

5 B

6 D

7 C

8 D & C

9 B & (D & C)

10 ∼ A ⊃ [Β & (D & C)]

1, 4 ⊃E

2, 5 ⊃E

3, 6 ⊃E

6, 7 &I

5, 8 &I

4–9 ⊃I

300

e. Derive: A

1 (∼ A ⊃ ∼ B)& (∼ B ⊃ B) Assumption

e. Derive: A

1 (∼ A ⊃ ∼ B)& (∼ B ⊃ B) Assumption

2 ∼ A A /∼ E

3 ∼ A ⊃ ∼ B 1 &E

4 ∼ B 2, 3 ⊃E

5 ∼ B ⊃ B 1 &E

6 B 4, 5 ⊃E

7 A 2–6 ∼ E

300

e. Derive: E TRIPLE BAR O

1 (E ⊃ T) & (T ⊃ O) Assumption

2 O ⊃ E Assumption

e. Derive: E TRIPLE BAR O

1 (E ⊃ T) & (T ⊃ O) Assumption

2 O ⊃ E Assumption

3 E A / I

4 E ⊃ T 1 &E

5 T 3, 4 ⊃E

6 T ⊃ O 1 &E

7 O 5, 6 ⊃E

8 O A / I

9 E 2, 8 ⊃E

10 E O 3–7, 8–9 I

300

g. Derive: (A ⊃ B) ⊃ [(C ⊃ A) ⊃ (C ⊃ B)]

1 A ⊃ B A / ⊃I

2 C ⊃ A A / ⊃I

3 C A / ⊃I

4 A 2, 3 ⊃E

5 B 1, 4 ⊃E

6 C ⊃ B 3–5 ⊃I

7 (C ⊃ A) ⊃ (C ⊃ B) 2–6 ⊃I

8 (A ⊃ B) ⊃ [(C ⊃ A) ⊃ (C ⊃ B)] 1–7 ⊃I

400

label the missing justifications

k. Derive: (A & B) ⊃ (C & D)

1 (B & A) ⊃ (D & C) Assumption

2 A & B A / ⊃I

3 B

4 A

5 B & A

6 D & C

7 C

8 D

9 C & D

10 (A & B) ⊃ (C & D)

2 &E

3, 4 &I

1, 5 ⊃E

6 &E

7, 8 &I

2–9 ⊃I

400

i. Derive: A ⊃ (B ⊃ C)

1 (A & B) ⊃ C Assumption

i. Derive: A ⊃ (B ⊃ C)

1 (A & B) ⊃ C Assumption

2 A A / ⊃Ι

3 B A / ⊃Ι

4 A & B 2, 3 & I

5 C 1, 4 ⊃E

6 B ⊃ C 3–5 ⊃I

7 A ⊃ (B ⊃ C) 2–6 ⊃I

400

Derive: D ⊃ B

1 A ⊃ C Assumption

2 (∼ A ∨ C) ⊃ (D ⊃ B) Assumption

Derive: D ⊃ B

1 A ⊃ C Assumption

2 (∼ A ∨ C) ⊃ (D ⊃ B) Assumption

3 ∼ (∼ A ∨ C) A /∼ E

4 A A /∼ I

5 C 1, 4 ⊃E

6 ∼ A ∨ C 5 ∨I

7 ∼ (∼ A ∨ C) 3 R

8 ∼ A 4–7 ∼I

9 ∼ A ∨ C 8 ∨I

10 ∼ (∼ A ∨ C) 3 R

11 ∼ A ∨ C 3, 10 ∼ E

12 D ⊃ B 2, 11 ⊃E

400

k. Derive: A ⊃ [B ⊃ (A ⊃ B)]

1 A A / ⊃I

2 B A / ⊃I

3 A A / ⊃I

4 B 2 R

5 A ⊃ B 3–4 ⊃I

6 B ⊃ (A ⊃ B) 2–5 ⊃I

7 A ⊃ [B ⊃ (A ⊃ B)] 1–6 ⊃I

499

label the missing justifications

Derive: (A & B) ⊃ E

1 A ⊃ C Assumption

2 B ⊃ D Assumption

3 (C & D) ⊃ E Assumption

4 A & B A / ⊃I

5 A

6 B

7 C

8 D

9 C & D

10 E

11 (A & B) ⊃ E

4 &E

1, 5 ⊃E

2, 6 ⊃E

7, 8 &I

3, 9 ⊃E

4–10 ⊃1

499

c. Derive: D ∨ E

1 E ∨ D Assumption

c. Derive: D ∨ E

1 E ∨ D Assumption

2 E A / ∨E

3 D ∨ E 2 ∨I

4 D A / ∨E

5 D ∨ E 4 ∨ I

6 D ∨ E 1, 2–3, 4–5 ∨E

499

Derive: B ∨ ∼ C

1 A ∨ (B & C) Assumption

2 C ⊃ ∼ A Assumption

Derive: B ∨ ∼ C

1 A ∨ (B & C) Assumption

2 C ⊃ ∼ A Assumption

3 A A / ∨E

4 C A /∼ I

5 ∼ A 2, 4 ⊃E

6 A 3 R

7 ∼ C 4–6 ∼ 1

8 B ∨ ∼ C 7 ∨I

9 B & C A / ∨E

10 B 9 &E

11 B ∨ ∼ C 10 ∨I

12 B ∨ ∼ C 1, 3–8, 9–11 ∨E

499

Derive: A ∨ (B & C)

1 (A ∨ B) & (A ∨ C) Assumption

2 A ∨ B 1 &E

3 A A / ∨E

4 A ∨ (B & C) 3 ∨I

5 B A / ∨E

6 A ∨ C 1 &E

7 A A / ∨E

8 A ∨ (B & C) 7 ∨I

9 C A / ∨E

10 B & C 5, 9 &I

11 A ∨ (B & C) 10 ∨I

12 A ∨ (B & C) 6, 7–8, 9–11 ∨E

13 A ∨ (B & C) 2, 3–4, 5–12 ∨E