Quantification = statement about an open sentence?

Question

The book I'm reading is talking about quantification being a method to convert open sentences into statements. From what I can see this method boils down to making a statement about the solution set of an open sentence P (x) over a domain S? Other than "for every x which is an element of S, P(x)" or "there exists atleast one x in S, such that P(x)", I could quantify it in many other ways like "there are atleast 2 distinct x in S, such that P(x)" or "There are less than 24 unique x in S such that P(x)" right?

Maybe those statements are less interesting than the universal and existential quantifier, but aren't they equally valid?

On a side note let P(x) be the condition required to be in a set P, and let S be some universal set which P is a subset of. Then would the statement "there exists an x in S such that P(x)" be equivalent to stating P is nonempty, and the statement "for every x in S, P(x)" be equivalent to stating P=S?

Answer 1

What André Nicolas said in his comment is correct. For example, you can say "at least $2$ objects satisfy $P$" by "$\exists x,y\ ( x \ne y \land P(x) \land P(y) )$". Similarly you can say "at most $3$ objects satisfy $P$" by "$\exists x,y,z\ ( \forall w\ ( P(w) \to w=x \lor w=y \lor w=z ) )$". Note however that you cannot say "finitely many objects satisfy $P$" in first-order logic, and by "cannot" I mean that this fact of impossibility can be proven.

Answer 2

Yes, you can invent different quantifiers. The interesting thing is precisely that they may be able to express concepts that can't be expressed in first-order logic.

Beyond examples like "there are finitely many objects" or "there are uncountable many objects", you can look at infinitary logics, in which formulas may have infinitely many free variables. At that point, you can consider infinitary quantifications such as $(\exists x_0 \exists x_1 \exists x_2 \dots \varphi)$ or $(\forall x_0 \forall x_1 \forall x_2 \dots \varphi)$. There are also game quantifiers such as $(\exists x_0 \forall x_1 \exists x_2 \forall x_3 \dots \varphi)$ or $(\forall x_0 \exists x_1 \forall x_2 \exists x_3 \dots \varphi),$ which state the existence of a winning strategy in a certain game.