What is "$\exists x \in S , \phi(x)$" shorthand for?

Question

I recently learned that "$\forall x \in S, \phi(x)$" is shorthand for "$\forall x \big(x \in S \rightarrow \phi(x)\big)$". Does the same idea apply for the shorthand "$\exists x \in S, \phi(x)$"?

i.e. is "$\exists x \in S, \phi(x)$" logically equivalent to "$\exists x \big(x \in S \rightarrow \phi(x)\big)$"?

I am somewhat tempted to think that it may actually denote "$\exists x\big( x \in S \land \phi(x)\big )$" but am not positive! Any insight is greatly appreciated. Cheers~

Answer 1

$\exists x{\in}S~.\phi(x)$ is intended to be read as "There is something, call it $x$, which is in $S$ and satisfies $\phi(x)$," and so is indeed synonymous with $\exists x~.(x\in S\wedge\phi(x))$

Moreover, we want quantifier duality to hold in restricted domains, so will need $\neg\forall x{\in}S~.\neg\phi(x)$ and $\exists x{\in}S~.\phi(x)$ to be logical equivalences.

Now because $\neg \forall x{\in}S~.\neg\phi(x)$ is synonymous with $\neg\forall x~.(x\in S\to \neg\phi(x))$, which by quantifier duality is equivalent to $\exists x~.\neg(x\in S\to\neg\phi(x))$, and thus with $\exists x~.(x\in S\wedge\phi(x))$, therefore we shall want this to be synonymous with $\exists x{\in}S~.\phi(x)$ .

Answer 2

I am somewhat tempted to think that it may actually denote "$\exists x\big( x \in S \land \phi(x)\big )$"

Yes, that's what it means.

i.e. is "$\exists x \in S, \phi(x)$" logically equivalent to "$\exists x \big(x \in S \rightarrow \phi(x)\big)$"?

That would not be very useful, because $x \in S \rightarrow \phi(x)$ is true for every $x \notin S$, no matter what $\phi$ is, and so $\exists x \big(x \in S \rightarrow \phi(x)\big)$ is true for any $S$ that is not the whole universe.