7.6.xa. Exercise answers

These answers use the rules RUG, SB, SC, and MCR rather than RUP and RUC. Derivations using the latter rules can be constructed from them by replacing each use of one of the first four rules by a series of three steps as shown in the following table:

basic rule   alternative approach
using RUP and RUC
RUG   RUC, UG, CP
SB   RUP, UI, MPP
SC   RUP, UI, MTT
MCR   RUP, UI, RC
1. a.
│∀x ∀y Rxy a:1
│(∀x: Rxx) Gx a:3
├─
1 UI │∀y Ray a:2
2 UI │Raa (3)
3 SB │Ga (4)
│●
├─
4 QED │Ga
  b.
│(∀x: Fx) Gx a:3
├─
│ⓐ
│││Fa (3)
││├─
3 SB │││Ga (4)
│││●
││├─
4 QED │││Ga 2
│├─
2 CP ││Fa → Ga 1
├─
1 UG │∀x (Fx → Gx)
 
│∀x (Fx → Gx) a:2
├─
│ⓐ
││Fa (3)
│├─
2 UI ││Fa → Ga 3
3 MPP ││Ga (4)
││●
│├─
4 QED ││Ga 1
├─
1 RUG │(∀x: Fx) Gx
  c.
│Fa (2)
├─
│ⓑ
││b = a a—b
│├─
││●
│├─
2 QED= ││Fb 1
├─
1 RUG │(∀x: x = a) Fx
 
│(∀x: x = a) Fx a:2
├─
││¬ Fa (2)
│├─
2 SC ││¬ a = a (3)
││●
│├─
3 DC ││⊥ 1
├─
1 IP │Fa
  d.
│∀x ∀y (Rxy → ¬ Ryx) a:5
├─
│ⓐ
││ⓑ
│││¬ a = b
││├─
││││Rab ∧ Rba 4
│││├─
4 Ext ││││Rab (7)
4 Ext ││││Rba (8)
5 UI ││││∀y (Ray → ¬ Rya) b:6
6 UI ││││Rab → ¬ Rba 7
7 MPP ││││¬ Rba (8)
││││●
│││├─
8 Nc ││││⊥ 3
││├─
3 RAA │││¬ (Rab ∧ Rba) 2
│├─
2 RUG ││(∀y: ¬ a = y) ¬ (Ray ∧ Rya) 1
├─
1 UG │∀x (∀y: ¬ x = y) ¬ (Rxy ∧ Ryx)
  e.
│∀x (∀y: ¬ x = y) ¬ (Rxy ∧ Ryx) a:5
│∀x ¬ Rxx a:8
├─
│ⓐ
││ⓑ
││││Rab (6),(9)
│││├─
│││││Rba (6)
││││├─
5 UI │││││(∀y:¬ a = y) ¬(Ray ∧ Rya) b:7
6 Adj │││││Rab ∧ Rba X,(7)
7 SC │││││a = b a—b
8 UI │││││¬ Raa (9)
│││││●
││││├─
9 Nc= │││││⊥ 4
│││├─
4 RAA ││││¬ Rba 3
││├─
3 CP │││Rab → ¬ Rba 2
│├─
2 UG ││∀y (Ray → ¬ Rya) 1
├─
1 UG │∀x ∀y (Rxy → ¬ Ryx)
  f.
│(∀x: Px) (∀y: Py) (∀z: Pz ∧ Lzx) Lyz b:6,a:10
├─
│ⓐ
││Pa (8), (10)
│├─
││ⓑ
│││Pb (6)
││├─
││││Lab (8)
│││├─
││││ⓒ
│││││Pc (11)
││││├─
│││││ⓓ
││││││Pd (7), (12)
│││││├─
6 SB ││││││(∀y: Py) (∀z: Pz ∧ Lzb) Lyz d:7
7 SB ││││││(∀z: Pz ∧ Lzb) Ldz a:9
8 Adj ││││││Pa ∧ Lab X, (9)
9 SB ││││││Lda (12)
10 SB ││││││(∀y: Py) (∀z: Pz ∧ Lza) Lyz c:11
11 SB ││││││(∀z: Pz ∧ Lza) Lcz d:13
12 Adj ││││││Pd ∧ Lda X, (13)
13 SB ││││││Lcd (14)
││││││●
│││││├─
14 QED ││││││Lcd 5
││││├─
5 RUG │││││(∀w: Pw) Lcw 4
│││├─
4 RUG ││││(∀z: Pz) (∀w: Pw) Lzw 3
││├─
3 CP │││Lab → (∀z: Pz) (∀w: Pw) Lzw 2
│├─
2 RUG ││(∀y: Py) (Lay → (∀z: Pz) (∀w: Pw) Lzw) 1
├─
1 RUG │(∀x:Px) (∀y:Py) (Lxy → (∀z:Pz) (∀w:Pw) Lzw)
  g.
│∀x (∀y: gx = y) Fy ha:2
├─
│ⓐ
2 UI ││(∀y: g(ha) = y) Fy g(ha):4
3 EC ││g(ha) = g(ha) X, (4)
4 SB ││F(g(ha)) (5)
││●
│├─
5 QED ││F(g(ha)) 1
├─
1 UG │∀x F(g(hx))
  h.
│∀x ∀y Rxy b:5
│(∀x: ∀y Ryx) (Fx → Gx) a:3
├─
│ⓐ
││Fa (8)
│├─
│││¬ Ga (9)
││├─
││││ⓑ
5 UI │││││∀y Rby a:6
6 UI │││││Rba (7)
│││││●
││││├─
7 QED │││││Rba 4
│││├─
4 UG ││││∀y Rya 3
│││
││││Fa → Ga 8
│││├─
8 MPP ││││Ga (9)
││││●
│││├─
9 Nc ││││⊥ 3
││├─
3 MCR │││⊥ 2
│├─
2 IP ││Ga 1
├─
1 RUG │(∀x: Fx) Gx

 

  i.
│(∀x: Rax) Sax c:9
│Pa ∧ ∀x ¬ Sax 1
│(∀x: Px ∧ ∀y ¬ Rxy) ∀z Fxz a:4
├─
1 Ext │Pa (6)
1 Ext │∀x ¬ Sax c:8
│ⓑ
│││¬ Fab
││├─
│││││●
││││├─
6 QED │││││Pa 5
││││
│││││ⓒ
8 UI ││││││¬ Sac (9)
9 SC ││││││¬ Rac (10)
││││││●
│││││├─
10 QED ││││││¬ Rac 7
││││├─
7 UG │││││∀y ¬ Ray 5
│││├─
5 Cnj ││││Pa ∧ ∀y ¬ Ray 4
│││
││││∀z Faz b:9
│││├─
││││Fab
││││●
│││├─
││││⊥ 4
││├─
4 MCR │││⊥ 3
│├─
3 IP ││Fab 2
├─
2 UG │∀z Faz

 

2. 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:3
│(∀x: Sx) Dx a:2
│∀x (Cx → Px) a:4
├─
│ⓐ
││Sa (2)
│├─
2 SB ││Da (3)
3 SB ││Ca (5)
4 UI ││Ca → Pa
5 MPP ││Pa (6)
││●
│├─
6 QED ││Pa 1
├─
1 RUG │(∀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:3
│(∀x: Sx) Dx a:2
│∀x (Rx → Cx) a:4
├─
│ⓐ
││Sa (2)
│├─
2 SB ││Da (3)
3 SB ││¬ Ca (5)
4 UI ││Ra → Ca 5
5 MTT ││¬ Ra (6)
││●
│├─
6 QED ││¬ Ra 1
├─
1 RUG │(∀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:3
│(∀x: Dx) Mx a:2
│∀x (Rx → Cx) a:4
├─
│ⓐ
││¬ Ma (2)
│├─
2 SC ││¬ Da (3)
3 SB ││¬ Ca (5)
4 UI ││Ra → Ca 5
5 MTT ││¬ Ra (6)
││●
│├─
6 QED ││¬ Ra 1
├─
1 RUG │(∀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:7
│(∀x: Sx) Dx a:3
│∀x (Cx → Px) a:4
├─
│ⓐ
││Sa ∧ ¬ Pa 2
│├─
2 Ext ││Sa (3)
2 Ext ││¬ Pa (5)
3 SB ││Da (6)
4 UI ││Ca → Pa 5
5 MTT ││¬ Ca (6)
6 Adj ││Da ∧ ¬ Ca X, (7)
7 SB ││La (8)
││●
│├─
8 QED ││La 1
├─
1 RUG │(∀x: Sx ∧ ¬ Px) Lx

 

  e. Everyone watched every snake
Every cobra is a snake
Everyone watched every cobra
   
│(∀x: Px) (∀y: Sy) Wxy a:3
│(∀x: Cx) Sx b:4
├─
│ⓐ
││Pa (3)
│├─
││ⓑ
│││Cb (4)
││├─
3 SB │││(∀y: Sy) Way b:5
4 SB │││Sb (5)
5 SB │││Wab (6)
│││●
││├─
6 QED │││Wab 2
│├─
2 RUG ││(∀y: Cy) Way 1
├─
1 RUG │(∀x: Px) (∀y: Cy) Wxy

 

  f. No one watched every snake
Every snake is a reptile
No one watched every reptile
   
│(∀x: Px) ¬ (∀y: Sy) Wxy a:2
│(∀x: Sx) Rx b:6
├─
│ⓐ
││Pa (2)
│├─
2 SB ││¬ (∀y: Sy) Way 4
││
│││(∀y: Ry) Way b:7
││├─
││││ⓑ
│││││Sb (6)
││││├─
6 SB │││││Rb (7)
7 SB │││││Wab (8)
│││││●
││││├─
8 QED │││││Wab 5
│││├─
5 RUG ││││(∀y: Sy) Way 4
││├─
4 CR │││⊥ 3
│├─
3 RAA ││¬ (∀y: Ry) Way 1
├─
1 RUG │(∀x: Px) ¬ (∀y: Ry) Wxy

 

  g. No one watched any snake
Every cobra is a snake
No one watched any cobra
   
│(∀y: Sy) (∀x: Px) ¬ Wxy a:3
│(∀x: Cx) Sx a:2
├─
│ⓐ
││Ca (2)
│├─
2 SB ││Sa (3)
3 SB ││(∀x: Px) ¬ Wxa (4)
││●
│├─
4 QED ││(∀x: Px) ¬ Wxa 1
├─
1 RUG │(∀y: Cy) (∀x: Px) ¬ Wxy

 

  h. Everyone who likes every snake was pleased
Every snake is a reptile
Everyone who likes every reptile was pleased
   
│(∀x: Px ∧ (∀y: Sy) Lxy) Dx a:4
│(∀x: Sx) Rx b:8
├─
│ⓐ
││Pa ∧ (∀y: Ry) Lay 2
│├─
2 Ext ││Pa (5)
2 Ext ││(∀y: Ry) Lay b:9
││
│││¬ Da (4)
││├─
4 SC │││¬ (Pa ∧ (∀y: Sy) Lay) 5
5 MPT │││¬ (∀y: Sy) Lay 6
│││
││││ⓑ
│││││Sb (8)
││││├─
8 SB │││││Rb (9)
9 SB │││││Lab (10)
│││││●
││││├─
10 QED │││││Lab 7
│││├─
7 RUG ││││(∀y: Sy) Lay 6
││├─
6 RC │││⊥ 3
│├─
3 IP ││Da 1
├─
1 RUG │(∀x: Px ∧ (∀y: Ry) Lxy) Dx

 

  i. Everyone who likes a snake was pleased
Every cobra is a snake
Everyone who likes a cobra was pleased
   
│(∀x: Sx) (∀y: Py ∧ Lyx) Dy a:3
│(∀x: Cx) Sx a:2
├─
│ⓐ
││Ca (2)
│├─
2 SB ││Sa (3)
3 SB ││(∀y: Py ∧ Lya) Dy (4)
││●
│├─
4 QED ││(∀y: Py ∧ Lya) Dy 1
├─
1 RUG │(∀x: Cx) (∀y: Py ∧ Lyx) Dy
Glen Helman 01 Aug 2004