Basically, we need quantifiers not so much for making inferences but for having enough expressive "capability" in our language.
For simplicity, I assume the context of classical first-order logic, where the existential quantifier $\exists$ can be defined as an abbreviation for $\lnot \forall \lnot$.
If we are working into formalized arithmetic (e.g.first-order logic with Peano's Axioms), we may stay with the convention that free variables are implicitly universally quantified.
So, the Axiom about the successor function :
$x = y \rightarrow S(x) = S(y)$
can be interpreted as $\forall x \forall y ( ...)$, like in algebra when we write : $(x+y)^2 = x^2 + 2xy +y^2$.
The problem is that not all the relevant "facts" can be rightly expressed with universal quantifiers upfront.
Take the formula : $x = 0$; if we stay with the above convention, its meaning must be : "all numbers are equal to $0$", that is false.
If we negate it, we get $\lnot x = 0$, which, following the convention, must be read as : "all number are different from $0$". But this is false also !
We need a sign for "restricting" the context (the scope) of the quantifier (this wonderful discovery was made by Frege) in order to have :
$\forall x (x = 0)$
Now, its negation will be :
$\lnot \forall x (x = 0)$
and this time we have the intended meaning : "not all numbers are equal to $0$", that is true.
This example is taken form S.C.Kleene, Introduction to Metamathematics (1952).