Quantifiers generalized from logical connectives I am aware of existence of the "generalized quantifiers" as seen here: https://en.wikipedia.org/wiki/Generalized_quantifier to mean "sets of sets". Universal quantifier is a set containing the universe and existential one is a family of all the subsets of the universe but the empty set. You can create more quantifiers by inventing new sets of sets. I got it, more or less.
What I'd like to know is whether anyone pursued other way to generalize the quantifiers - as infinite logical connectives. You can think of universal quantifier as inifite and operaton (can't you?):
$\forall_{x\in X}: p(x) \approx \bigwedge_{x\in X} p(x) \approx p(x_1) \wedge p(x_2)\wedge\dots\wedge p(x_i)\wedge\dots$
Similarly for existential quantifier as inifite or:
$\exists_{x\in X}: p(x) \approx \bigvee_{x\in X} p(x) \approx p(x_1) \vee p(x_2)\vee\dots\vee p(x_i)\vee\dots$
Has anyone found a way to do the same with other logical connectives?
I know that all other connectives can be constructed from these two, so perhaps there is nothing to discover/invent here, but still... I find it interesting to try to evaluate the logical value for
$?_{x\in X}: p(x) \approx \Leftrightarrow_{x\in X} p(x) \approx p(x_1) \leftrightarrow p(x_2)\leftrightarrow\dots\leftrightarrow p(x_i)\leftrightarrow\dots$
which seems to depend on the (possibly infite) parity of the domain $X$ (whatever that means, if anything, as pointed out in the comments).
 A: 
What I'd like to know is whether anyone pursued other way to generalize the quantifiers - as infinite logical connectives.

Yes. In fact, this is of primary importance to the study of infinitary logics. In some theories, the infinite conjunction and disjunction are exactly the universal and existential quantifiers, respectively.

Has anyone found a way to do the same with other logical connectives?

Incidentally, I dealt with this exact problem a few months ago. Here's what I came up with:
Let $U$ be a domain and $C:\mathcal{P}(U)\not\to U$ the choice function on $U$ such that $C(E)\in E$ for all $E$ in the domain of $C$. For each predicate $\phi:U\to\Bbb{B}$, define the operator $\Pi^\phi_C$ according to the following:

*

*for any set $E\subseteq U$,

$$\Pi^\phi_C(E)\equiv\bigg{(}\phi(C(E))\iff\Pi^\phi_C(E\setminus \{C(E)\})\bigg{)}$$


*for any expression $P$,

$$P\iff\Pi^\phi_C(\emptyset)\equiv P$$
Suppose now that $U=\{x_n:n\in\Bbb{N}\}$ and for any $E\subseteq U$,
$$C(E)=x_i\quad \text{iff}\quad i=\min\{j\in\Bbb{N}:x_j\in E\}$$
Then:
$$\begin{align}
\Pi^\phi_C(U)&\equiv\phi(x_0)\iff\Pi^\phi_C(U\setminus\{x_0\})\\
&\equiv\phi(x_0)\iff\phi(x_1)\iff\Pi^\phi_C(U\setminus\{x_0,x_1\})\\
&\cdots\\
&\equiv\phi(x_0)\iff\phi(x_1)\iff\phi(x_2)\iff\cdots
\end{align}$$
This provides a finite means of expressing the new quantifier in terms of the logical connective $\iff$ - independently of "parity" of the set, as you put it.
That being said, I would be extremely cautious using this technique. The existence of the choice function is not provable in every theory. Even then, the evaluation of $\Pi^\phi_C(X)$ is another matter entirely. It is not always possible to evaluate $\Pi^\phi_C(E)$ for all $\phi$, $C$, and $E$.
In particular, if $E$ is uncountable, then there is no finite procedure to determine $\Pi^\phi_C(E)$ for arbitrary $C$.
Addendum
For each predicate $\phi$, let $\mathcal{U}_\phi$ be an ultrafilter over the powerset of the domain $U$ such that the ultrapower $\Bbb{B}^U/\mathcal{U}\cong\Bbb{B}$, then define one of the equivalence classes to be the set of infinite bi-implications whose valuation is "true."
This can be done with or without choice, but the selection of the "correct" ultrafilter can be difficult when the order on $U$ is not apparent. In most cases, it can be shown, informally, that some ultrafilter exists; but this information is not necessarily helpful for evaluating specific infinite statements.
