why can only universal statements be vacuously true? So I'm new to logic and im getting really confused on definitions namely that of universal and existential quantifiers and why vacuous truths only apply to statements with a universal quantifier $(\forall)$ and not for existential quantifier $(\exists)$.
I read that the 2 statements
(1) $\forall x\in A\,P(x)$
(2) $\forall x(x\in A\implies P(x))$
Are logically equivalent but why cant we say the same thing about the existential quantifier namely why cant we say that statements $(1)$,$(2)$ can be used.
(1) $\exists x\in A\,P(x)$
(2) $\exists x(x\in A\implies P(x))$
So why do we say that only universal or conditional statements are vacuously true?.
 A: The formula $\exists x \in A\;P(x)$ stands for $\exists x(x \in A \land P(x))$. This is because we want the following chain of equivalences to hold:
$$
\begin{align*}
\exists x \in A\;P(x) &\equiv \lnot \forall x \in A\;\lnot P(x) \\
 &\equiv \lnot \forall x(x \in A \implies \lnot P(x)) \\
 &\equiv \exists x(x \in A \land P(x))
\end{align*}
$$
where the first equivalence is the desired De Morgan law, the second follows from the definition as you give it and the third follows because to falsify $\phi \implies \psi$ you have to make $\phi$ true and $\psi$ false.
$\exists x(x \in A \implies P(x))$ would be trivially true (unless $A$ is the entire universe of discourse) since for $x \not\in A$, $x \in A \implies \phi$ is true for any $\phi$.
The analogue of the vacuously true universal: $\forall x \in \emptyset\;P(x)$ is the vacuously false existential: $\exists x \in \emptyset\;P(x)$.
A: As I understand it, the implication $A\to B$ is said to be vacuously true if $A$ is false. We can see this from lines 3 and 4 of the truth table for implication:

$\neg A \to (A\to B)$ is also a tautology:

This can also be proven using a form of natural deduction:

Vacuous truth doesn't necessarily have anything to do with quantifiers or empty sets, though it is used, for example, to show that the empty set is unique, i.e. that all empty sets are equal. It is also used in proofs by cases when one or more of the cases being considered are proven or assumed to be false.
