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)