Reference Request: Had those notations for quantification been explored before?

Question

The following is simply an endeavour to explore alternative ways of writing quantifier notation for fragments of first\higher order logic.

1. The arrow notation:

$\uparrow \! x, \phi \iff \forall x \, \phi \\ \downarrow \! x , \phi \iff \exists x \, \phi\\ {\overset {\large \downarrow} .} \, x, \phi \iff \exists ! x \, \phi$

Examples:

Extensionality: $\uparrow \! x , \uparrow \! y, (\uparrow \! z , (z \in x \leftrightarrow z \in y) \to x=y)$

Foundation: $\downarrow \! x , (x \in y ) \to , \downarrow \! x, (x \in y \land \neg \downarrow \! z , (z \in x \land z \in y))$

Union: $\uparrow \! A, \downarrow \! x, \uparrow \! y , (y \in x \leftrightarrow , \downarrow \! z , (z \in A \land y \in z))$

Power: $\uparrow \! A, \downarrow \! x, \uparrow \! y , (y \in x \leftrightarrow , \uparrow \! z , (z \in y \to y \in A))$

Separation: $\uparrow \! A, \downarrow \! x, \uparrow \! y , (y \in x \leftrightarrow y \in A \land \phi(y,A))$

Replacement: $\uparrow \! A, \ (\uparrow \! x \in A, {\overset {\large \downarrow} .} \, y, \phi(x,y) \to \\\downarrow \! B, \uparrow \!y , (y \in B \leftrightarrow \, \, \downarrow \! x, (x \in A \land \phi(x,y))))$

Infinity: $\downarrow \! x, : \varnothing \in x \ \land \uparrow \! y , (y \in x \to \{y\} \in x)$

Choice: $\uparrow \! X, \ [ \uparrow \!m, \uparrow \! n , (m \in X \land n \in X\to \, \, \neg \downarrow \! l, : l \in m \land l \in n) \\\to, \downarrow \! C, \uparrow \! x \in X, {\overset {\large \downarrow} .} \, c, : c \in C \land c \in x] $

I'm still of the opinion that this way of writing quantifiers is the simplest appropriate way of doing it. The pitfull is that it doesn't easily illustrate a spectrum of quantification concepts as the following alternative does:

2. The circle notation:

${\LARGE \bullet} x \, \phi \iff \forall x \, \phi$

$^{\large \odot} x \, \phi \iff \neg \, {\LARGE \bullet } x \, \neg \phi$

Examples:

Choice: ${\LARGE \bullet} \! X \ [ {\LARGE \bullet} m, {\LARGE \bullet} n \, (m \in X \land n \in X\to \, \, \neg \, ^{\large \odot} l: l \in m \land l \in n) \to \\ ^{\large \odot}\! C, {\LARGE \bullet} x \in X, ^{\large \odot !} \! c: c \in C \land c \in x] $

Infinity: $^{\large \odot}\! x: \varnothing \in x \ \land {\LARGE \bullet} y \in x \, (\{y\} \in x)$

Replacement: ${\LARGE \bullet} A \ ({\LARGE \bullet} x \in A, ^{\large \odot !} \! y: \phi(x,y) \to \\^{\large \odot} \! B, {\LARGE \bullet} y \, (y \in B \leftrightarrow \, ^{\large \odot} x \in A: \phi(x,y)))$

This notation have the blessing of showing a spectrum of quantification, as follows:

$^\bigcirc x \, \phi \iff {\LARGE \bullet} x \, \neg \phi$; read as: Non x

$^{\Huge \circ}\kern-0.930 em ^{\huge \bullet} \! x \, \phi \iff |\{x \mid \phi \}| > |\{x \mid \neg \phi\}|$; read as: Most $x$

$^{\Huge \circ}\kern-0.780em ^{\Large \bullet}\! x \, \phi \iff |\{x \mid \phi \}| < |\{x \mid \neg \phi\}|$; read as: Few $x$

$^ {\large \unicode{x29bf}} x \, \phi \iff |\{x \mid \phi \}| = |\{x \mid \neg \phi\}|$; read has Half $x$

$^{\large \odot}\kern-0.546em ^{\Large .} x \, \phi \iff ^{\large \odot} x ^{\large \odot} y: x \neq y \land \phi(x) \land \phi(y)$; read as: At least two $x$

$^{\large \odot !} x \iff ^{{\large \odot }^1} x \iff ^{\large \odot} l: {\LARGE \bullet} x \, ( \phi(x) \leftrightarrow x=l)$; read as: there exists a unique $x$

$^{{\large \odot }^n} x \iff ^{\large \odot} l_1,..,^{\large \odot} \! l_n: \overset {\rightarrow} {l_{i \leq n} \neq l_{j \leq n}} \land \\ {\LARGE \bullet} x \, ( \phi(x) \leftrightarrow (x=l_1 \lor..\lor x=l_n))$

;read as: there exists exactly $n$ $x$.

We can also adopt another version of circle notation which is somewhat the opposite of the above, where the white replaces black in the above representation, so for example Choice would be written as:

Choice: $^\bigcirc \! X \ [ ^\bigcirc m, ^\bigcirc \! n \, (m \in X \land n \in X\to \, \, \neg \, ^{\large \odot} l: l \in m \land l \in n) \to \\ ^{\large \odot} C, ^\bigcirc \! x \in X, ^{\large \odot !} \! c: c \in C \land c \in x] $

I find the circles more demonstrative, while the arrows simpler. We can of course modify the arrow notation as to demonstrate the spectrum of quantification concepts, but it would be cumbersome. I think the circle notation is less arbitrary than the arrows. To me both notations seem to be less arbitrary than the customary $\forall, \exists$ language bound notation, and more into capturing aspects of the notion of quantification.

Has any of these notations been attempted before for symbolising quantification?

What are the main drawbacks (other than it not being the convention) for these notations?