Ever since I was a young student I have felt doubts about the traditional $(\forall x)$-expression: starting a statement with such an irrational lack of focus doesn't seems reasonable! I mean, all $x$ $-$ /mind explosion/, is so much that it is absolutely impossible to overview. Here $x$ can be anything: a bacteria that penetrated the cell membrane of some urmetazoan cell and become a part of a process that ended with the creation of the complex mitochondria organelle; a proton that once was a part of a cloud of dust swirled around in a nebulous and now is a part of a neuron in your brain; a sign invented by some being somewhere in space; a class of something; or even a number...
A lot of mathematics is written with the alternative, and logically much more limited, expression $\forall x\in A$, which in my opinion is absolutely unobjectionable. Is it then correct to claim that the mathematical statement $\forall x\in A:P(x)$ is founded on the corresponding first order logical statement $(\forall x)(x\in A\rightarrow P(x))$?
Otherwise, I claim, most mathematical theories substantially are founded on propositional logic. Except for the relation $\in$..!
Statements as $(a\in A)$ and $(a\in A\rightarrow P(a))$, where $a$ is a free variable, are substantially structured statements in propositional logic, and with the simple semantic convention:
$(1)\quad(\forall x\in A:P(x))\leftrightarrow(a\in A\rightarrow P(a))$
which induce the complementary rule:
$(2)\quad(\exists x\in A:P(x))\leftrightarrow(\neg\forall x\in A:\neg P(x))\leftrightarrow(a\in A\wedge P(a))$, due to $(1)$.
any mathematical statement can be transformed to such a propositional statement.
Conjecture: any structured compound propositional statement can be consistently interpreted as a statement with quantifiers. And different but correct propositional manipulations gives equivalent quantified statements $-$ if all free variables are restricted as members of a given set.
Note: all bounded variables must be consequently transformed to free variables and conversely. E.g. when a free variable is transformed into a bounded, all occurrences of that variable must be transformed simultaneously into the same quantifier expression of bounded variables.
Example: if $\mathbb N_m=\{x\in\mathbb N|x>m\}$ and $\oplus$ is exclusive or:
$1\oplus(\varepsilon\in\mathbb R_+)\oplus(\varepsilon\in\mathbb R_+)(m\in\mathbb N) \oplus(\varepsilon\in\mathbb R_+)(m\in\mathbb N)(n\in\mathbb N_m)\oplus$ $(n\in\mathbb N_m)(|x-x_n|<\varepsilon)\leftrightarrow$ $(\varepsilon\notin\mathbb R_+)\oplus(\varepsilon\in\mathbb R_+)(m\in\mathbb N) (1+(n\in\mathbb N_m))\oplus(n\in\mathbb N_m)(|x-x_n|<\varepsilon)\leftrightarrow$ $(\varepsilon\notin\mathbb R_+)\oplus(\varepsilon\in\mathbb R_+)(m\in\mathbb N) (n\notin\mathbb N_m)\oplus(n\in\mathbb N_m)(|x-x_n|<\varepsilon)\leftrightarrow$ $(\varepsilon\notin\mathbb R_+)\vee (m\in\mathbb N\wedge n\notin\mathbb N_m)\vee (m\in\mathbb N\wedge|x-x_n|<\varepsilon)\leftrightarrow$ $\varepsilon\in\mathbb R_+\rightarrow m\in\mathbb N\wedge(n\in\mathbb N_m\rightarrow |x-x_n|<\varepsilon)\leftrightarrow$ $\forall\epsilon\in\mathbb R_+\exists k\in\mathbb N:n\in\mathbb N_{k}\rightarrow |x-x_n|<\epsilon\leftrightarrow \displaystyle \lim_{i\rightarrow\infty}x_i=x$.
Question: are there any adequate reason (except traditional or authority reasons) why the restricted semantic quantifiers in mathematics must be defined by the corresponding, much more general, first order logical notation? For example a counter example of the conjecture?