TITLE: Were logical quantifiers not primarily motivated by infinite domains of discourse?
QUESTION [6 upvotes]: There is this quote by Hermann Weyl:
"… 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
$\exists x\ \phi(x)\equiv\phi(a)\lor\phi(b)\lor\phi(c)$
and
$\forall x\ \phi(x)\equiv\phi(a)\land\phi(b)\land\phi(c).$
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.

REPLY [2 votes]: In Logic, quantifiers are undefined, i.e., primitive concepts.  They have to satisfy axioms.  But their interpretation is as open and variable as the interpretation of the primitives of Boolean algebra.
Henkin quantifiers are an example of non-standard interpretations of quantifiers that obey all the usual axioms for quantifiers, but disagree with our intuition of there exists or for all.  See Alonzo Church, Mathematical Logic, in my opinion the only satisfactory introductory textbook.
Historically, Hilbert and his pupils had a lot of trouble axiomatising the quantifier. Several of their first publications were wrong: this history is explained in Church.
There is a difference in esprit between logicians and mathematicians.  Set theory is a branch of mathematics, and shares the same esprit.  Goedel never really understood Russell.  For Russell, there could only be one logic, and it was about the real world, Russell's interest in logic grew out of his own attempt to axiomatise Physics.  That is why when Relativity and later Quantum Theory came, he and Whitehead abandoned the Principia unfinished and he wrote about Relativity instead, and Whitehead developed a rival gravitational theory to Einstein's (for quite a while there were rivals to the Einsteinians: the Whiteheadians, of which there still remained at Queen's Univ., when I was there,  a 90 year old professor in the Physics Dept. ... still lecturing and doing research...). When Whitehead retired he threw out his copy of Principia and it was picked up by a grad student from the hallway....
That said, I will now rest some, since it should be obvious that one's motivation for introducing quantifiers will depend on whether one comes from Physics or Logic or Mathematics.  I suppose Cauchy with his definition of limit was early: clearly this answers your question «yes, uncountable».  R was his universe.  Frege, even more so: his motivation was to define a cardinal number as an equivalence class of sets.  But with the theory of types, things finally get interesting: Physics will tell us whether there are uncountably many, countably many, or finitely many objects.  There are no sets, and so there are only countably many propositions or formulas built on top of the objects (higher type), so second-order quantifiers only deal with countably many instantiations.  Furthermore, how many proper names can there be in a formal language? Only countably many..... people seem to have lost interest in this problem.