|
b.
|
| │A ∨ (B ∧ C) | 3,8 |
| ├─ | |
| │││¬ A | (5) |
| ││├─ | |
| ││││A | (5) |
| │││├─ | |
| │││││¬ B | |
| ││││├─ | |
| │││││● | |
| ││││├─ | |
5 Nc | │││││⊥ | 4 |
| │││├─ | |
4 IP | ││││B | 3 |
| │││ | |
| ││││B ∧ C | |
| │││├─ | |
6 Ext | ││││B | 7 |
6 Ext | ││││C | |
| ││││● | |
| │││├─ | |
7 QED | ││││B | 3 |
| ││├─ | |
3 PC | │││B | 2 |
| │├─ | |
2 PE | ││A ∨ B | 1 |
| │ | |
| │││A | |
| ││├─ | |
| ││││¬ C | |
| │││├─ | |
| ││││○ | A, ¬ C ⊭ ⊥ |
| │││├─ | |
| ││││⊥ | 9 |
| ││├─ | |
9 IP | │││C | 8 |
| ││ | |
| │││B ∧ C | 10 |
| ││├─ | |
10 Ext | │││B | |
10 Ext | │││C | 11 |
| │││● | |
| ││├─ | |
11 QED | │││C | 8 |
| │├─ | |
8 PC | ││C | 1 |
| ├─ | |
1 Cnj | │(A ∨ B) ∧ C | |
|
|
Since entailment fails in one direction, equivalence must fail, so a second derivation for entailment in the other direction need not be pursued; but that entailment does hold, as is shown below.
| │(A ∨ B) ∧ C | 1 |
| ├─ | |
1 Ext | │A ∨ B | 2 |
1 Ext | │C | (8) |
| │ | |
| ││A | (4) |
| │├─ | |
| │││¬ (B ∧ C) | |
| ││├─ | |
| │││● | |
| ││├─ | |
4 QED | │││A | 3 |
| │├─ | |
3 PE | ││A ∨ (B ∧ C) | 2 |
| │ | |
| ││B | (7) |
| │├─ | |
| │││¬ A | |
| ││├─ | |
| ││││● | |
| │││├─ | |
7 QED | ││││B | 6 |
| │││ | |
| ││││● | |
| │││├─ | |
8 QED | ││││C | 6 |
| ││├─ | |
6 Cnj | │││B ∧ C | 5 |
| │├─ | |
5 PE | ││A ∨ (B ∧ C) | 2 |
| ├─ | |
2 PC | │A ∨ (B ∧ C) | |
Each of the following divides the one open gap:
A | B | C | A | ∨ | (B | ∧ | C) | / | (A | ∨ | B) | ∧ | C |
T | T | F | | Ⓣ | | F | | | | T | | Ⓕ |
T | F | F | | Ⓣ | | F | | | | T | | Ⓕ |
|