Phi 270
Fall 2013
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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
   
ABC(A∧B)→C/A→C
TFFFⓉⒻ
   
│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
  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)
   
ABCA→C/(A→B)∧(B→C)
TFTⓉFⒻT
FTFⓉTⒻF

The two rows represent counterexamples lurking in the first and second gap, respectively.

   
│(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
  d.
││(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
   

The following is a second approach to this derivation; it uses one of the forms of Wk for the conditional:

││(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 and2
│    Indianapolis
│Tom won’t visit Ursula without going through 3
│    Indianapolis
├─
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 television1
│     or we won’t have wide public exposure
│If we spend a bundle on television, we’ll go2
│    into debt
│Either we have wide public exposure4
│     or our contributions will dry up
│We’ll go into debt if our contributions dry up6
│     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 debt1
│
││We won’t have wide public exposure(4)
│├─
4 MTP││Our contributions will dry up(5)
5 Adj││Our contributions dry upX,(6)
││     and we won’t have large reserves
6 MPP││We’ll go into debt(7)
││●
│├─
7 QED││We’ll go into debt1
├─
1 PC│We’ll go into debt
  g.
│If Adams supports the plan, it will go though3
│     provided Brown doesn’t oppose it
│Brown won’t oppose the plan5
│     if either Collins or Davis supports it
├─
││Both Adams and Davis will support the plan2
│├─
2 Ext││Adams will support the plan(3)
2 Ext││Davis will support the plan(4)
3 MPP││The plan will go though provided Brown6
││    doesn’t oppose it
4 Wk││Either Collins or Davis will support the planX,(5)
5 MPP││Brown won’t oppose the plan(6)
6 MPP││The plan will go through(7)
││●
│├─
7 QED││The plan will go through1
├─
1 CP│The plan will go through
│     if both Adams and Davis support it
Glen Helman 01 Aug 2013