7.5.xa. Exercise answers

1.
      instance for a   instance for b   instance for c
a. ∀x Fx   Fa   Fb   Fc
b. ∀y Fy   Fa   Fb   Fc
c. ∀x Rxa   Raa   Rba   Rca
d. ∀x Saxb   Saab   Sabb   Sacb
e. ∀x ∀y Rxy   ∀y Ray   ∀y Rby   ∀y Rcy
f. ∀x (Fx → Gx)   Fa → Ga   Fb → Gb   Fc → Gc
g. ∀x (Fx → Gd)   Fa → Gd   Fb → Gd   Fc → Gd
h. ∀x (Fx → ∀y Rxy)   Fa → ∀y Ray   Fb → ∀y Rby   Fc → ∀y Rcy
i. ∀x (Fx → ∀x Rxx)   Fa → ∀x Rxx   Fb → ∀x Rxx   Fc → ∀x Rxx
2. a.
│∀x Fx a:1
│∀x (Fx → Gx) a:2
├─
1 UI │Fa (3)
2 UI │Fa → Ga 3
3 MPP │Ga (4)
│●
├─
4 QED │Ga
  b.
│∀x (Fx ∧ Gx) a:1, b:3
├─
1 UI │Fa ∧ Ga 2
2 Ext │Fa (5)
2 Ext │Ga
3 UI │Fb ∧ Gb 4
4 Ext │Fb
4 Ext │Gb (5)
5 Adj │Fa ∧ Gb (6)
│●
├─
6 QED │Fa ∧ Gb
  c.
│∀x Rxa b:1
│∀x (Rbx → Gx) a:2
├─
1 UI │Rba (3)
2 UI │Rba → Ga 3
3 MPP │Ga (4)
│●
├─
4 QED │Ga
  d.
│∀x ∀y Rxy a:1
│∀x (Rxx → Gx) a:3
├─
1 UI │∀y Ray a:2
2 UI │Raa (4)
3 UI │Raa → Ga 4
4 MPP │Ga (5)
│●
├─
5 QED │Ga
  e.
│∀x ∀y Rxy a:1, b:3
├─
1 UI │∀y Ray b:2
2 UI │Rab (5)
3 UI │∀y Rby b:4, a:6
4 UI │Rbb (5)
5 Adj │Rab ∧ Rbb X, (7)
6 UI │Rba (7)
7 Adj │(Rab ∧ Rbb) ∧ Rba X, (8)
│●
├─
8 QED │(Rab ∧ Rbb) ∧ Rba
  f.
│∀x Fx a:2
│∀x (Fx → Gx) a:3
├─
│ⓐ
2 UI ││Fa (4)
3 UI ││Fa → Ga 4
4 MPP ││Ga (5)
││●
│├─
5 QED ││Ga 1
├─
1 UG │∀x Gx
  g.
│∀x (Fx ∧ Gx) a:3,b:7
├─
││ⓐ
3 UI │││Fa ∧ Ga 4
4 Ext │││Fa
4 Ext │││Ga (5)
│││●
││├─
5 QED │││Fa 2
│├─
2 UG ││∀x Fx 1
│
││ⓑ
7 UI │││Fb ∧ Gb 8
8 Ext │││Fb
8 Ext │││Gb (9)
│││●
││├─
9 QED │││Gb 6
│├─
6 UG ││∀x Gx 1
├─
1 Cnj │∀x Fx ∧ ∀x Gx
 
│∀x Fx ∧ ∀x Gx 1
├─
1 Ext │∀x Fx a:3
1 Ext │∀x Gx a:4
│ⓐ
3 UI ││Fa (5)
4 UI ││Ga (5)
5 Adj ││Fa ∧ Ga X, (6)
││●
│├─
6 QED ││Fa ∧ Ga 2
├─
2 UG │∀x (Fx ∧ Gx)
   

Reusing the term a as the independent term of the second general argument of the derivation on the left would have caused no logical problems since the two gaps are separate arguments boxed off from on another; however, we will hold to the simplest interpretation of the scope line and not allow terms flagging scope line to appear anywhere outside their indicated scope.

  h.
│Fa (3)
├─
│ⓑ
│││b = a a—b
││├─
│││●
││├─
3 QED= │││Fb 2
│├─
2 CP ││b = a → Fb 1
├─
1 UG │∀x (x = a → Fx)
 
│∀x (x = a → Fx) a:2
├─
││¬ Fa (3)
│├─
2 UI ││a = a → Fa 3
3 MTT ││¬ a = a (4)
││●
│├─
4 DC ││⊥ 1
├─
1 IP │Fa
  i.
│∀x ∀y Rxy b:2
├─
│ⓑ
2 UI ││∀y Rby a:3
3 UI ││Rba (4)
││●
│├─
4 QED ││Rba 1
├─
1 UG │∀y Rya

Here the term a cannot be used as the independent term of the general argument because it already appears in the conclusion.

  j.
│∀x ∀y (Rxy → ¬ Ryx) a:3
├─
│ⓐ
│││Raa (5), (6)
││├─
3 UI │││∀y (Ray → ¬ Rya) a:4
4 UI │││Raa → ¬ Raa 5
5 MPP │││¬ Raa (6)
│││●
││├─
6 Nc │││⊥ 2
│├─
2 RAA ││¬ Raa 1
├─
1 UG │∀x ¬ Rxx
  k.
│∀x ∀y (gx = y → Fy) ha:2
├─
│ⓐ
2 UI ││∀y (g(ha) = y → Fy) g(ha):3
3 UI ││g(ha) = g(ha) → F(g(ha)) 5
4 EC ││g(ha) = g(ha) X, (5)
5 MPP ││F(g(ha)) (6)
││●
│├─
6 QED ││F(g(ha)) 1
├─
1 UG │∀x F(g(hx))
3. a. Every road sign was colored
Every stop sign was a road sign
If anything was colored, it was painted
Every stop sign was painted
   
│∀x (Dx → Cx) a:5
│∀x (Sx → Dx) a:3
│∀x (Cx → Px) a:7
├─
│ⓐ
│││Sa (4)
││├─
3 UI │││Sa → Da 4
4 MPP │││Da (6)
5 UI │││Da → Ca 6
6 MPP │││Ca (8)
7 UI │││Ca → Pa 8
8 MPP │││Pa (9)
│││●
││├─
9 QED │││Pa 2
│├─
2 CP ││Sa → Pa 1
├─
1 UG │∀x (Sx → Px)
  b. No road sign was colored
Every stop sign was a road sign
If anything was red, it was colored
No stop sign was red
   
│∀x (Dx → ¬ Cx) a:5
│∀x (Sx → Dx) a:3
│∀x (Rx → Cx) a:7
├─
│ⓐ
│││Sa (4)
││├─
3 UI │││Sa → Da 4
4 MPP │││Da (6)
5 UI │││Da → ¬ Ca 6
6 MPP │││¬ Ca (8)
7 UI │││Ra → Ca 8
8 MTT │││¬ Ra (9)
│││●
││├─
9 QED │││¬ Ra 2
│├─
2 CP ││Sa → ¬ Ra 1
├─
1 UG │∀x (Sx → ¬ Rx)

 

  c. Only road signs were colored
Every road sign was a traffic marker
If anything was red, it was colored
Only traffic markers were red
   
│∀x (¬ Dx → ¬ Cx) a:5
│∀x (Dx → Mx) a:3
│∀x (Rx → Cx) a:7
├─
│ⓐ
│││¬ Ma (4)
││├─
3 UI │││Da → Ma 4
4 MTT │││¬ Da (6)
5 UI │││¬ Da → ¬ Ca 6
6 MPP │││¬ Ca (8)
7 UI │││Ra → Ca 8
8 MTT │││¬ Ra (9)
│││●
││├─
9 QED │││¬ Ra 2
│├─
2 CP ││¬ Ma → ¬ Ra 1
├─
1 UG │∀x (¬ Mx → ¬ Rx)

 

  d. Among road signs, all except colored ones were replaced
Every stop sign was a road sign
If anything was colored, it was painted
Among stop signs, all except painted ones were replaced
   
│∀x ((Dx ∧ ¬ Cx) → Lx) a:8
│∀x (Sx → Dx) a:4
│∀x (Cx → Px) a:6
├─
│ⓐ
│││Sa ∧ ¬ Pa 3
││├─
3 Ext │││Sa (5)
3 Ext │││¬ Pa (7)
4 UI │││Sa → Da 5
5 MPP │││Da (9)
6 UI │││Ca → Pa 7
7 MTT │││¬ Ca (9)
8 UI │││(Da ∧ ¬ Ca) → La 10
9 Adj │││Da ∧ ¬ Ca X, (10)
10 MPP │││La (11)
│││●
││├─
11 QED │││La 2
│├─
2 CP ││(Sa ∧ ¬ Pa) → La 1
├─
1 UG │∀x ((Sx ∧ ¬ Px) → Lx)
Glen Helman 25 Aug 2009