7.8.xa. Exercise answers

1.
│∀x ¬ ∀y ¬ Rxy a:3
├─
│ⓐ
│││Raa
││├─
3 UI │││¬ ∀y ¬ Ray 4
│││
││││││Raa
│││││├─
││││││○ Raa ⊭ ⊥
│││││├─
││││││⊥ 6
││││├─
6 RAA │││││¬ Raa 5
││││
││││ⓑ
│││││(unfinished)
││││├─
│││││¬ Rab 5
│││├─
5 UG+ ││││∀y ¬ Ray 4
││├─
4 CR │││⊥ 2
│├─
2 RAA ││¬ Raa 1
├─
1 UG │∀x ¬ Rxx
 

From the 1st open gap (the 2nd is not fully developed).

2.
│∀x ¬ ∀y Rxy a:2, b:8
├─
││∀x Rxa a:3, b:9
│├─
2 UI ││¬ ∀y Ray 4
3 UI ││Raa (6)
││
││││●
│││├─
6 QED ││││Raa 5
│││
│││ⓑ
│││││¬ Rab
││││├─
8 UI │││││¬ ∀y Rby 10
9 UI │││││Rba (12)
│││││
│││││││●
││││││├─
12 QED │││││││Rba 11
││││││
││││││││¬ Rbb
│││││││├─
││││││││○ Raa,¬ Rab,Rba,¬ Rbb ⊭ ⊥
│││││││├─
││││││││⊥ 13
││││││├─
13 IP │││││││Rbb 11
││││││
││││││ⓒ
│││││││(unfinished)
││││││├─
│││││││Rbc 11
│││││├─
11 UG+ ││││││∀y Rby 10
││││├─
10 CR │││││⊥ 7
│││├─
7 IP ││││Rab 5
││├─
5 UG+ │││∀y Ray 4
│├─
4 CR ││⊥ 1
├─
1 RAA │¬ ∀x Rxa
 

From the 1st open gap (the 2nd is not fully developed)

3.
│∀x ¬ ∀y Rxy a:3, c:8
├─
│││Raa (6)
││├─
3 UI │││¬ ∀y Ray 4
│││
│││││●
││││├─
6 QED │││││Raa 5
││││
││││ⓒ
││││││¬ Rac
│││││├─
8 UI ││││││¬ ∀y Rcy 9
││││││
│││││││││¬ Rca
││││││││├─
│││││││││○ Raa,¬ Rac,¬ Rca ⊭ ⊥
││││││││├─
│││││││││⊥ 11
│││││││├─
11 IP ││││││││Rca 10
│││││││
│││││││││¬ Rcc
││││││││├─
│││││││││○ Raa,¬ Rac,¬ Rcc ⊭ ⊥
││││││││├─
│││││││││⊥ 12
│││││││├─
12 IP ││││││││Rcc 10
│││││││
│││││││ⓓ
││││││││(unfinished)
│││││││├─
││││││││Rcd 10
││││││├─
10 UG+ │││││││∀y Rcy 9
│││││├─
9 CR ││││││⊥ 7
││││├─
7 IP │││││Rac 5
│││├─
5 UG+ ││││∀y Ray 4
││├─
4 CR │││⊥ 2
│├─
2 RAA ││¬ Raa 1
│
│ⓑ
││(unfinished)
│├─
││¬ Rab 1
├─
1 UG+ │∀x ¬ Rax
 

From the 1st and 2nd open gaps (the 3rd and 4th are not fully developed)

Glen Helman 19 Jul 2012