3.4.xa. Exercise answers

1. a.
│¬ B
├─
││A ∧ ¬ B 2
│├─
2 Ext ││A
2 Ext ││¬ B
││○ A, ¬ B ⇏ ⊥
│├─
││⊥ 1
├─
1 RAA │¬ (A ∧ ¬ B)
A B ¬ B / ¬ (A ∧ ¬ B)
T F Ⓣ Ⓕ T T
b.
│¬ (A ∧ B) 3,8
├─
│││A (5)
││├─
│││││●
││││├─
5 QED │││││A 4
││││
││││││¬ B
│││││├─
││││││○ A, ¬ B ⇏ ⊥
│││││├─
││││││⊥ 6
││││├─
6 IP │││││B 4
│││├─
4 Cnj ││││A ∧ B 3
││├─
3 CR │││⊥ 2
│├─
2 RAA ││¬ A 1
│
│││B (11)
││├─
││││││¬ A
│││││├─
││││││○ ¬ A, B ⇏ ⊥
│││││├─
││││││⊥ 10
││││├─
10 IP │││││A 9
││││
│││││●
││││├─
11 QED │││││B 9
│││├─
9 Cnj ││││A ∧ B 8
││├─
8 CR │││⊥ 7
│├─
7 RAA ││¬ B 1
├─
1 Cnj │¬ A ∧ ¬ B
A B ¬ (A ∧ B) / ¬ A ∧ ¬ B
T F Ⓣ F F Ⓕ T
F T Ⓣ F T Ⓕ F

The first row is an interpretation that divides the first gap; and the second row is an interpretation that divides the second gap.

  c.
│¬ (A ∧ B) 3
│¬ (B ∧ C) 7
├─
││A ∧ C 2
│├─
2 Ext ││A
2 Ext ││C (10)
││
││││●
│││├─
5 QED ││││A 4
│││
│││││¬ B
││││├─
││││││││¬ B
│││││││├─
││││││││○ A, ¬ B, C ⇏ ⊥
│││││││├─
││││││││⊥ 9
││││││├─
9 IP │││││││B 8
││││││
│││││││●
││││││├─
10 QED │││││││C 8
│││││├─
8 Cnj ││││││B ∧ C 7
││││├─
7 CR │││││⊥ 6
│││├─
6 IP ││││B 4
││├─
4 Cnj │││A ∧ B 3
│├─
3 CR ││⊥ 1
├─
1 RAA │¬ (A ∧ C)
A B C ¬ (A ∧ B), ¬ (B ∧ C) /¬ (A ∧ C)
T F T Ⓣ F Ⓣ F Ⓕ T
2. a.
│¬ (A ∧ ¬ B) 2
├─
││¬ B (5)
│├─
│││││¬ A
││││├─
│││││○ ¬ A, ¬ B ⇏ ⊥
││││├─
│││││⊥ 4
│││├─
4 IP ││││A 3
│││
││││●
│││├─
5 QED ││││¬ B 3
││├─
3 Cnj │││A ∧ ¬ B 2
│├─
2 CR ││⊥ 1
├─
1 RAA │B
A B ¬ (A ∧ ¬ B) / B
F F Ⓣ F T Ⓕ
b.
│¬ (A ∧ B) 3
├─
││B ∧ A 2
│├─
2 Ext ││B (6)
2 Ext ││A (5)
││
││││●
│││├─
5 QED ││││A 4
│││
││││●
│││├─
6 QED ││││B 4
││├─
4 Cnj │││A ∧ B 3
│├─
3 CR ││⊥ 1
├─
1 RAA │¬ (B ∧ A)
c.
│¬ (A ∧ ¬ B) 3
├─
││B ∧ ¬ A 2
│├─
2 Ext ││B
2 Ext ││¬ A
││
│││││¬ A
││││├─
│││││○ ¬ A, B ⇏ ⊥
││││├─
│││││⊥ 5
│││├─
5 IP ││││A 4
│││
│││││B
││││├─
│││││○ ¬ A, B ⇏ ⊥
││││├─
│││││⊥ 6
│││├─
6 IP ││││¬ B 4
││├─
4 Cnj │││A ∧ ¬ B 3
│├─
3 CR ││⊥ 1
├─
1 RAA │¬ (B ∧ ¬ A)
A B ¬ (A ∧ ¬ B) / ¬ (B ∧ ¬ A)
F T Ⓣ F F Ⓕ T T
d.
│¬ (A ∧ B) 3,8
│¬ (B ∧ C) 11
│B (6),(10),(14)
├─
│││A (5)
││├─
│││││●
││││├─
5 QED │││││A 4
││││
│││││●
││││├─
6 QED │││││B 4
│││├─
4 Cnj ││││A ∧ B 3
││├─
3 CR │││⊥ 2
│├─
2 RAA ││¬ A 1
│
│││C (15)
││├─
││││││¬ A
│││││├─
││││││││●
│││││││├─
14 QED ││││││││B 13
│││││││
││││││││●
│││││││├─
15 QED ││││││││C 13
││││││├─
13 Cnj │││││││B ∧ C 12
│││││├─
12 CR ││││││⊥ 11
││││├─
11 IP │││││A 9
││││
│││││●
││││├─
10 QED │││││B 9
│││├─
9 Cnj ││││A ∧ B 8
││├─
8 CR │││⊥ 7
│├─
7 RAA ││¬ C 1
├─
1 Cnj │¬ A ∧ ¬ C
│¬ (A ∧ B) 3
│¬ (B ∧ C) 8
│B (6),(10)
├─
│││A (5)
││├─
│││││●
││││├─
5 QED │││││A 4
││││
│││││●
││││├─
6 QED │││││B 4
│││├─
4 Cnj ││││A ∧ B 3
││├─
3 CR │││⊥ 2
│├─
2 RAA ││¬ A 1
│
│││C (11)
││├─
│││││●
││││├─
10 QED │││││B 9
││││
│││││●
││││├─
11 QED │││││C 9
│││├─
9 Cnj ││││B ∧ C 8
││├─
8 CR │││⊥ 7
│├─
7 RAA ││¬ C 1
├─
1 Cnj │¬ A ∧ ¬ C

The derivation on the left exploits resources in their order of appearance; while the one above chooses, at stage 8, the resource that is most closely connected with other resources of the gap in which it is exploited. Notice that derivation on the left is eventually led to exploit the same resource to the same effect.

e.
│¬ (A ∧ ¬ (B ∧ ¬ (C ∧ ¬ D))) 3
├─
││A ∧ ¬ (B ∧ D) 2
│├─
2 Ext ││A (5)
2 Ext ││¬ (B ∧ D) 8
││
││││●
│││├─
5 QED ││││A 4
│││
│││││B ∧ ¬ (C ∧ ¬ D) 7
││││├─
7 Ext │││││B (10)
7 Ext │││││¬ (C ∧ ¬ D) 12
│││││
│││││││●
││││││├─
10 QED │││││││B 9
││││││
││││││││¬ D (15)
│││││││├─
│││││││││││¬ C
││││││││││├─
│││││││││││○ A, B, ¬ C, ¬ D ⇏ ⊥
││││││││││├─
│││││││││││⊥ 14
│││││││││├─
14 IP ││││││││││C 13
│││││││││
││││││││││●
│││││││││├─
15 QED ││││││││││¬ D 13
││││││││├─
13 │││││││││C ∧ ¬ D 12
│││││││├─
12 CR ││││││││⊥ 11
││││││├─
11 IP │││││││D 9
│││││├─
9 Cnj ││││││B ∧ D 8
││││├─
8 CR │││││⊥ 6
│││├─
6 RAA ││││¬ (B ∧ ¬ (C ∧ ¬ D)) 4
││├─
4 Cnj │││A ∧ ¬ (B ∧ ¬ (C ∧ ¬ D)) 3
│├─
3 CR ││⊥ 1
├─
1 RAA │¬ (A ∧ ¬ (B ∧ D))
A B C D ¬ (A ∧ ¬ (B ∧ ¬ (C ∧ ¬ D))) / ¬ (A ∧ ¬ (B ∧ D))
T T F F Ⓣ F F T T F T Ⓕ T T F
Glen Helman 28 Aug 2008