I'm trying to prove whether "all loves all" (everyone loves everyone) is a tautology or not using the tree method. While this statement shouldn't be a logical truth, my tree closes (tree setup is $\forall x \forall y Lxy$ , $\neg \forall x \forall y Lxy$ ). Where did I go wrong?
1) ∀x∀y Lxy (premise) 2) ¬∀x∀y Lxy (negation) ∃x ¬∀y Lxy from 2) ¬∀y Lay from 3) ∃y ¬Lay from 4) ¬Lab from 5) ∀y Lay from 1) Lab from 7) (Tree closes)
1) ¬∀x∀y Lxy (negation) ∃x ¬∀y Lxy from 1) ¬∀y Lay from 2) ∃y ¬Lay from 3) ¬Lab from 4)