5.4.xa. Exercise answers

1. a.
│A → B 2
├─
││A (2)
│├─
2 MPP ││B (3)
││●
│├─
3 QED ││B 1
├─
1 PE │¬ A ∨ B
 
│¬ A ∨ B 2
├─
││A (2)
│├─
2 MTP ││B (3)
││●
│├─
3 QED ││B 1
├─
1 CP │A → B
  b.
│(A ∧ B) → C 3
├─
││A (4)
│├─
│││¬ C (3)
││├─
3 MTT │││¬ (A ∧ B) 4
4 MPT │││¬ B
│││○ A, ¬ C, ¬ B ⊭ ⊥
││├─
│││⊥ 2
│├─
2 IP ││C 1
├─
1 CP │A → C
│A → C 3
├─
││A ∧ B 2
│├─
2 Ext ││A (3)
2 Ext ││B
3 MPP ││C (4)
││●
│├─
4 QED ││C 1
├─
1 CP │(A ∧ B) → C
ABC(A∧B)→C/A→C
TFFFⓉⒻ
  c.
│A → C 3,7
├─
│││A (3)
││├─
3 MPP │││C
│││
││││¬ B
│││├─
││││○ A, C, ¬ B ⊭ ⊥
│││├─
││││⊥ 4
││├─
4 IP │││B 2
│├─
2 CP ││A → B 1
│
│││B
││├─
││││¬ C (7)
│││├─
7 MTT ││││¬ A
││││○ B, ¬ C, ¬ A ⊭ ⊥
│││├─
││││⊥ 6
││├─
6 IP │││C 5
│├─
5 CP ││B → C 1
├─
1 Cnj │(A → B) ∧ (B → C)
│(A → B) ∧ (B → C) 1
├─
1 Ext │A → B 3
1 Ext │B → C 4
│
││A (3)
│├─
3 MPP ││B (4)
4 MPP ││C (5)
││●
│├─
5 QED ││C 2
├─
2 CP │A → C


ABCA→C/(A→B)∧(B→C)
TFTⓉFⒻT
FTFⓉTⒻF

The two rows divide the first and second gap, respectively.

  d.

The following are two approaches to this derivation, one without use of attachment rules and the other using one of the forms of Wk for the conditional.

 
││(A → B) → A 3
│├─
│││¬ A (3),(7)
││├─
3 MTT │││¬ (A → B)
│││
│││││A (7)
││││├─
││││││¬ B
│││││├─
││││││●
│││││├─
7 Nc ││││││⊥ 6
││││├─
6 IP │││││B 5
│││├─
5 CP ││││A → B 4
││├─
4 CR │││⊥ 2
│├─
2 IP ││A 1
├─
1 CP │((A → B) → A) → A
││(A → B) → A 4
│├─
│││¬ A (3),(5)
││├─
3 Wk │││A → B X,(4)
4 MPP │││A (5)
│││●
││├─
5 Nc │││⊥ 2
│├─
2 IP ││A 1
├─
1 CP │((A → B) → A) → A
2. a.
│(A ∧ B) → C 2
│(C ∨ D) → E 4
│A (1)
│B (1)
├─
1 Adj │A ∧ B X,(2)
2 MPP │C (3)
3 Wk │C ∨ D X,(4)
4 MPP │E (5)
│●
├─
5 QED │E
  b.
│(A ∨ ¬ B) → C 2
├─
││¬ C (2)
│├─
2 MTT ││¬ (A ∨ ¬ B) (5)
││
│││¬ B (4)
││├─
4 Wk │││A ∨ ¬ B X,(5)
│││●
││├─
5 Nc │││⊥ 3
│├─
3 IP ││B 1
├─
1 CP │¬ C → B
  c.
│¬ (A ∧ B) 2
│B ∨ C 3
│D → ¬ C
├─
││A (2)
│├─
2 MPT ││¬ B (3)
3 MTP ││C (4)
4 MTT ││¬ D (5)
││●
│├─
5 QED ││¬ D 1
├─
1 CP │A → ¬ D
  d.
│C → ¬ (A ∨ B) 3
│E ∨ ¬ (D ∧ ¬ C) 5
│D (4)
├─
││A (2)
│├─
2 Wk ││A ∨ B X,(3)
3 MTT ││¬ C (4)
4 Adj ││D ∧ ¬ C X,(5)
5 MTP ││E (6)
││●
│├─
6 QED ││E 1
├─
1 CP │A → E
  e.
│Tom will go through Chicago and visit Sue 1
│Tom won’t go through both Chicago and Indianapolis 2
│Tom won’t visit Ursula without going through Indianapolis 3
├─
1 Ext │Tom will go through Chicago (2)
1 Ext │Tom will visit Sue (4)
2 MPT │Tom won’t go through Indianapolis (3)
3 MPT │Tom won’t visit Ursula (4)
4 Adj │Tom will visit Sue but not Ursula X,(5)
│●
├─
5 QED │Tom will visit Sue but not Ursula
  f.
│Either we spend a bundle on television 1
│     or we won’t have wide public exposure
│If we spend a bundle on television, we’ll go into debt 2
│Either we have wide public exposure 4
│     or our contributions will dry up
│We’ll go into debt if our contributions dry up 6
│     and we don’t have large reserves
│We won’t have large reserves (5)
├─
││We’ll spend a bundle on television (2)
│├─
2 MPP ││We’ll go into debt (3)
││●
│├─
3 QED ││We’ll go into debt 1
│
││We won’t have wide public exposure (4)
│├─
4 MTP ││Our contributions will dry up (5)
5 Adj ││Our contributions dry up X,(6)
││     and we won’t have large reserves
6 MPP ││We’ll go into debt (7)
││●
│├─
7 QED ││We’ll go into debt 1
├─
1 PC │We’ll go into debt
  g.
│If Adams supports the plan, 3
│     it will go though provided Brown doesn’t oppose it
│Brown won’t oppose the plan 5
│     if either Collins or Davis supports it
├─
││Both Adams and Davis will support the plan 2
│├─
2 Ext ││Adams will support the plan (3)
2 Ext ││Davis will support the plan (4)
3 MPP ││The plan will go though provided Brown doesn’t oppose it 6
4 Wk ││Either Collins or Davis will support the plan X,(5)
5 MPP ││Brown won’t oppose the plan (6)
6 MPP ││The plan will go through (7)
││●
│├─
7 QED ││The plan will go through 1
├─
1 CP │The plan will go through
│     if both Adams and Davis support it
Glen Helman 12 Oct 2011