I have been trying to proof if the following predicate logic formal is valid or not. Having trouble proving it because it has multiple variables.
(∀ ()) ∨ ∃((∀ (, ) ∨ ()) ⇒ ∃∀ (, ))
tried to simplify the formula:
(∀ ()) ∨ ∃((∀ (, ) ∨ ()) ⇒ ∃∀ (, ))
(∀ ()) ∨ ((∃ ∀ (, ) ∨ ∃ ()) ⇒ ∃∀ (, ))
(∀ ()) ∨ ( ∃ ∀ ((, ) ∨ () ⇒ ∃ (, )))
....
I have a feeling that it is non valid for having ∃ () in the implication, that shows that y can be false in (, ). Would appreciate it if someone can help me explain how to solve this kind of formula.
Well, it's a non-sequitur. How is $P$ related to either $Q$ or $R$. There is no way you can prove that sort of conditional where there is no connection between predicates unless the consequent is a tautology, or the antecedent is a contradiction. It's obviously invalid.