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!

I have to disagree with many of the answers here. There is a theme here that "humans are not computers, so it's easier for them to read words than symbols". But why stop at logical symbols? There are many symbols we do use in math in lieu of words. In earlier times, an equation like $x^3+3x=2$ would be written as "the thing cubed plus three times the thing equals two". Perhaps we should all go back to that?

Of course not. Mathematical symbols are here to make our lives easier, not harder - they do so by expressing ideas in a concise and easy-to-work-with form.

The question was also asked if we would like to read a proof that has no words and only symbols. But that is a strawman since nobody was suggesting this - instead, we wish to use a mixture of words and symbols according to what makes sense in the context, and that logical symbols will not be singled out for avoidance.

I can only speculate that the real reason behind this is that logic is, on one hand, ubiquitous in math, and on the other hand, formal mathematical logic was developed much later than algebra. So by the time mathematicians realized they actually can use symbols for logic, it has been widespread to use words for logical connectives, and symbols for the domain-specific subject matter.

And of course, traditions are hard to change - once everyone is used to one way, it's harder for them to parse statements written another way, and they find the traditional way prudent, so they continue the tradition in their own writing, enforce it in their journals, etc., and perpetuate the tradition. But it is important to understand that there is no real reason to eschew logical symbols - it's just a continuation of a tradition that resulted from a historical accident.

PS. $\forall$ and $\exists$ are rare in non-logic texts, but I do see $\implies$ and the like occasionally.

Proofs in textbooks are supposed to be readable for humans. Using logical symbols can make proofs significantly shorter but also significantly more confusing. The proofs written in natural language with actual words and sentences are much faster to read. Moreover, with words you can explain ideas, not just state truths. Mathematics is not only about finding all the true statements. Textbooks and shorter mathematical texts tell a story, and stories are hard to tell in symbols only.

A proof in words can also be difficult to understand if it never explains why things are done the way they are. If you only use symbols, you don't even have the chance to explain. A symbolic proof explained in words might work, but I find non-symbolic logic often easiest to follow in a proof.

Make a test if you will: Take a proof that is approximately one page long. Convert it to logical symbols so that no English word remains and give it to a friend to read. Can they understand what is going on?

Some things are better written in symbols, some in words, and the position of the borderline is a matter of taste. I believe most people would agree that writing mathematics with symbols only or without symbols of any kind is not a good idea.