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)
Edited:
1) ¬∀x∀y Lxy (negation)
∃x ¬∀y Lxy from 1)
¬∀y Lay from 2)
∃y ¬Lay from 3)
¬Lab from 4)
The edited version is correct ... And since you obtained a finished and open branch the original statement that 'all loves all' is not a tautology. And of course it isn't: a simple counterexample is a scenario where a does not love b .. Which is a counterexample provided by the tree.