The thing I'm trying to understand is why this problem is false? Isn't it the same?
2026-04-24 16:35:39.1777048539
Is $∀x((P(x)∨Q(x))) \to (∀x: (P(x) ∨ ∀x: Q(x)))$ correct.?
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The antecedent (the thing before the arrow) says everything is either $P$ or $Q$, the consequent (thing after the arrow) says either everything is $P$, or everything is $Q$. The former does not imply the latter. For instance, it could be that there are two things, one of which is $P$ and another is $Q$, but no thing is both $P$ and $Q$. In this case the antecedent is true, but the consequent is false while the antecedent is true.