"… classical logic was abstracted from the mathematics of finite sets and their subsets …. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory …." (Weyl, 1946)
Does this imply that at the invention of quantifiers "$\exists, \forall$", they were merely used for finite domains of discourse?
In that case, I think that would mean they can be viewed as merely abbreviating the connectives "$\land$, $\lor$". For example, if the domain conists of $3$ elements $a,b,c$, then merely
$\exists x\ \phi(x)\equiv\phi(a)\lor\phi(b)\lor\phi(c)$
$\forall x\ \phi(x)\equiv\phi(a)\land\phi(b)\land\phi(c).$
A motivation is to understand if I can think of quantifier rules as just induced by rules for connectives. For example, for finde domains, I figured that using only De Morgan's laws
$\neg\exists x\ \phi(x)=\neg((\phi(a)\lor\phi(b))\lor\phi(c))=\neg(\phi(a)\lor\phi(b))\land\neg\phi(c)=(\neg\phi(a)\land\neg\phi(b))\land\neg\phi(c)=\forall x\ \neg\phi(x).$
At a time were people haven't come up with the quantifiers, I guess they didn't even think about things like finitely axiomatizing arithmetic yet.
Since I came across the quote in the article discussing the controversy behind cardinalities:
Once we got quantifier for infinite domains (like the natural numbers), is there an additional obstacle for accepting uncountable domains?
I wonder about this point because if you introduce the quantifier, because you can't write down an infinitely long sentence, for example a $\forall$-claim about the element of $\mathbb Z$, then there is not so much difference anymore to the problem which arises with $\mathbb R$. Sure, they are not countable - but you can still name many of it's elements (even if you can't even specify some). You can only handle all of them via $\forall$, and that is true not matter what infinite cardinality you speak about.