Is forall and exists predicates?

Question

In a past thread it was mentioned that $x \in A$ is a predicate. I know $\exists x$ and $\forall x$ are quantifiers but are they also predicates themselves? What about when combined with "in" itself (or whatever this operator is called)? e.g. $\exists x \in A$ or $\forall x \in A$

Answer 1

No, the quantifiers are not predicates. Rather, combined with predicates, quantifiers can form claims. E.g. $\exists x \ x \in A$ would be the claim that there is some object $x$ that is an element of $A$

This is not the same as $\exists x \in A$ though, which is a restricted quantifier. You'd need to combine that quantifier with some predicate (or formula in general) to get a claim again. For example, $\exists x \in A \ P(x)$ is the claim that there is some element of $A$ that has property $P$.

Answer 2

Yes and no.

Let $X$ denote a set. A predicate on $X$ is basically a subset of $X$. A quantifier on $X$ is basically a collection of subsets of $X$. In particular, we can think of the existential quantifier as the collection of all non-empty subsets of $X$, and we can think of the universal quantifier as the collection $\{X\}$ with exactly one element, namely the predicate that is true everywhere. Under this interpretation, the meaning of $\forall x \in X: P(x)$ is roughly $P \in \forall_X$, which is equivalent to $P \in \{X\}$, which is equivalent to $P = X$, which is just saying that $P$ is true everywhere. Similarly, the meaning of $\exists x \in X:P(x)$ is roughly $P \in \exists_X$, where $\exists_X$ is the set of all predicates on $X$ that return True for at least one input. Of course, there's some extra subtleties when extra variables are around. For example, $\forall x \in X: P(x,y)$ means approximately $(x \mapsto P(x,y)) \in \forall_X$.

In light of all this, I tend to think that predicates and quantifiers are just different, and that in particular, neither generalizes the other. On the other hand, note that a quantifier on a set $X$ is just a predicate on the powerset of $X$. In some sense, every quantifier is a predicate, but just not on the same set.