Logic of simple statement concerning the empty set

I was curious whether the following statement is true or false.

Question. Let $P(x)$ be any statement pertaining to $x$. Is "$P(x)$ for all elements $x$ of the empty set is true" a true or a false statement?

I assume it is true because it’s vacuous. Am I right? Thanks :)

Solution 1:

As @Mauro ALLEGRANZA noted, "$P(x)$ for all elements $x$ of the empty set is true" is best translated as $\forall x(x\in\varnothing\to P(x))$, and so is true since the antecedent of the conditional is false, (hence, "trivially true").

By the way, for the question of @Aman Kushwaha, "There exists an $x\in\varnothing$ such that $\neg P(x)$ is true" could be translated as $\exists x(x\in\varnothing\wedge \neg P(x))$, and so is false since the conjunct $x\in\varnothing$ is false.