Why there is no sign of logic symbols in mathematical texts?

Many mathematicians, and I want to be in that number, believe that

Let us fix any $\epsilon>0$. It follows from the assumptions that there exists a positive number $\delta$ with the property that $1/x<\epsilon$ whenever $x>\delta$

is more elegant than

$(\forall \epsilon>0)(\exists \delta>0)(\forall x)(x > \delta \Rightarrow 1/x<\epsilon)$.

Book authors often want to write good books, carefully written, elegant and pleasant to read. Book authors often think of themselves as artists, or professional writers: if, as others said in their answers, good English grants both style and scientific quality to a book, why not use it?

Certain symbols are best used only in certain cases.

The most common place where such symbols are used, at least when not talking about the topic of logic, is likely on a blackboard. That's because they are used in addition with the spoken word. Then the symbols are an abbreviated language.

For example, the "therefore" symbol does not have any formal usage in mathematical logic, and I've hardly ever seen it in print, but it is great for a blackboard argument because the professor accompanies it with he spoken word, "therefore."

Logic symbols for print exist because sometimes we want to reason about logic itself.

The famous topologist James Munkres requires students in his MIT courses follow these guidelines on good mathematical style. Rule (7) reads

Don't use logical symbols at all. The symbols $\exists, \ni, \forall, \exists !, \vee, \wedge $ as well as the abbreviations s.t., w.r.t., are to be avoided in mathematical writing. In papers in logic, these symbols constitute part of the subject matter and are completely appropriate. In informal mathematical discourse, on blackboard or paper, they are often used as "parts of speech", in a sort of mathematical shorthand. However, they are not allowed by editors in formal mathematical writing.

Just as you wouldn't submit a history paper that is written partly in secretarial shorthand, don't submit a math paper written partly in mathematical shorthand!

Rule (8) adds

One exception is the use of the symbols $\Rightarrow$ (implies) and $\Leftarrow$ (is implied by) and $\Leftrightarrow$ (is equivalent to). One of course does not use these symbols as word-substitutes, any more than one uses $<$ or $+$ or $\cap$ as word-substitutes [e.g., "Consider the set of all numbers $< 1$" or "Consider the $\cap$ of the sets $A$ and $B$"].

But usage is allowed in phrases such as: "We show that (a) $\Rightarrow$ (b) $\Rightarrow$ (c)," or "To show (a) and (b) are equivalent, it suffices to show that (a) $\Rightarrow$ (b) and (b) $\Rightarrow$ (a)".

There is a reason why editors (at least those who are also mathematicians) enforce rule (7) strictly. Most mathematical readers find sentences in which this rule is violated quite unreadable, just as they find secretarial shorthand unreadable. They translate the sentence into the English language (or German, or French, or ...) mentally, before attempting to understand it. ...

Yes!