Why do other logical symbols except $\to$ and $\leftrightarrow$ not have corresponding meta symbols?

Question

The symbols $\to$ and $\leftrightarrow$ have corresponding meta symbols $\Rightarrow$ and $\iff$. But why other symbols like $\wedge, \vee,(,), \cdots $ do not have corresponding meta symbols? Is there are something logical reason?

Answer 1

The $\Rightarrow$ and $\Leftrightarrow$ assert important logical relationships:

$\varphi \Rightarrow \psi$ is a meta-logical assertion stating that the expression $\psi$ is a logical consequence of $\varphi$

$\varphi \Leftrightarrow \psi$ asserts that $\varphi$ and $\psi$ are logically equivalent

Clearly we cannot use $\to$ and $\leftrightarrow$ for these; those are mere truth-functional operators as part of the logic language itself; they don't assert any meta-logical relationship.

But what would be the role of a meta-logical symbol for 'and' or 'or'? What relationship would it express, or what claim would it make? In fact, if I say: "Sentence $\varphi$ AND sentence $\psi$", I am not even making a claim about $\varphi$ and $\psi$ at all, unlike $\varphi \Rightarrow \psi$ and $\varphi \Leftrightarrow \psi$