6.3.xa. Exercise answers
Some of the derivations below are given twice, once using only the basic identity rules EC, DC, QED=, and Nc= and a second time using MPP= and similar extensions for equations of other rules (see 6.3.3); either approach is entirely acceptable.
|
1.
|
| │Fa → Ga | 1 |
| │Fa | (1) |
| │a = b | a—b |
| ├─ | |
| 1 MPP | │Ga | (2) |
| │● | |
| ├─ | |
| 2 QED= | │Gb | |
|
|
2.
|
| │Fa → Ga | 2 |
| │Fb | (3) |
| │a = b | a—b |
| ├─ | |
| ││¬ Ga | (2) |
| │├─ | |
| 2 MTT | ││¬ Fa | (3) |
| ││● | |
| │├─ | |
| 3 Nc= | ││⊥ | 1 |
| ├─ | |
| 1 IP | │Ga | |
|
| |
| │Fa → Ga | 1 |
| │Fb | (1) |
| │a = b | a—b |
| ├─ | |
| 1 MPP= | │Ga | (2) |
| │● | |
| ├─ | |
| 2 QED | │Ga | |
|
|
3.
|
| │Fa ∧ a = gb | 1 |
| ├─ | |
| 1 Ext | │Fa | (4) |
| 1 Ext | │a = gb | a—gb, b, c, gc |
| │ | |
| ││¬ F(gc) | (4) |
| │├─ | |
| │││b = c | a—gb—gc, b—c |
| ││├─ | |
| │││● | |
| ││├─ | |
| 4 Nc= | │││⊥ | 3 |
| │├─ | |
| 3 RAA | ││¬ b = c | 2 |
| ├─ | |
| 2 CP | │¬ F(gc) → ¬ b = c | |
|
|
4.
|
| │Fa → G(fa) | 3 |
| │G(fb) → Hb | 5 |
| │a = b | a—b, fa—fb |
| ├─ | |
| ││Fb | (4) |
| │├─ | |
| │││¬ Ha | (7) |
| ││├─ | |
| ││││● | |
| │││├─ | |
| 4 QED= | ││││Fa | 3 |
| │││ | |
| ││││G(fa) | 6 |
| │││├─ | |
| │││││● | |
| ││││├─ | |
| 6 QED= | │││││G(fb) | 5 |
| ││││ | |
| │││││Hb | (7) |
| ││││├─ | |
| │││││● | |
| ││││├─ | |
| 7 Nc= | │││││⊥ | 5 |
| │││├─ | |
| 5 RC | ││││⊥ | 3 |
| ││├─ | |
| 3 RC | │││⊥ | 2 |
| │├─ | |
| 2 IP | ││Ha | 1 |
| ├─ | |
| 1 CP | │Fb → Ha | |
|
| |
| │Fa → G(fa) | 2 |
| │G(fb) → Hb | 3 |
| │a = b | a—b, fa—fb |
| ├─ | |
| ││Fb | (2) |
| │├─ | |
| 2 MPP= | ││G(fa) | (3) |
| 3 MPP= | ││Hb | (4) |
| ││● | |
| │├─ | |
| 4 QED= | ││Ha | 1 |
| ├─ | |
| 1 CP | │Fb → Ha | |
|
|
5.
|
| │fa = b | |
| │fc = d | a, b—fa, c, d—fc |
| ├─ | |
| ││a = c ∨ b = d | 2 |
| │├─ | |
| │││a = c | a—c, b—fa—fc—d |
| ││├─ | |
| │││● | |
| ││├─ | |
| 3 EC | │││fa = d | 2 |
| ││ | |
| │││b = d | a, fa—b—d—fc, c |
| ││├─ | |
| │││● | |
| ││├─ | |
| 4 EC | │││fa = d | 2 |
| │├─ | |
| 2 CP | ││fa = d | 1 |
| ├─ | |
| 1 CP | │(a = c ∨ b = d) → fa = d | |
|
|
6.
|
| │v = b | |
| │o = p | o—p, b—v |
| │¬ Fvi | (3) |
| ├─ | |
| ││Foi | (3) |
| │├─ | |
| │││b = p | o—p—b—v |
| ││├─ | |
| │││● | |
| ││├─ | |
| 3 Nc= | │││⊥ | 2 |
| │├─ | |
| 2 RAA | ││¬ b = p | 1 |
| ├─ | |
| 1 CP | │Foi → ¬ b = p | |
F: [ _ is from _ ]; v: the vice president; b: Barack Obama; c: Joe Biden; p: the president; t: Illinois
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