TITLE: About the position of "for all" quantifier
QUESTION [19 upvotes]: I'm not expert in logic, but as far as I know, quantifiers comes before the predicates they refer to. Still, if written in english, there are statements which sounds better when you don't put all the quantifiers before. For example, the definition of a sequence of functions $f_n:\mathbb{R}\to\mathbb{R}$ converging uniformly to some function $f:\mathbb{R}\to\mathbb{R}$: 

For all $\varepsilon > 0$, exists $n_0\in\mathbb{N}\ $ such that $\ n>n_0 \implies |f_n(x) - f(x)| < \varepsilon$ for all $x\in\mathbb{R}$

I have seen teachers writing this as 

$$\forall\varepsilon > 0,\ \ \exists n_0\in\mathbb{R}\ \  \text{ such that } \ \ n>n_0 \implies |f_n(x) - f(x)| < \varepsilon,\ \ \forall x\in\mathbb{R}$$

Maybe an even more simple example, from probability. It's not hard to find  something like

$$P[X_n=Y_n, \forall n\in\mathbb{N}]$$

where $(X_n)_{n\in\mathbb{N}}$ and $(Y_n)_{n\in\mathbb{N}}$ are collection of random variables. 
I understand the motivation: if you talk about this probability, you probably will say the probability of have $X_n=Y_n$ for all $n$, and not the probability of, for all $n$, have $X_n=Y_n$. The same reasoning applies for the statement about uniform convergence. That last for all quantifier will fit better at the final of the sentence if you have to say it in words.
So my question is: are this "informal" formulas just wrong (in the formal logic point of view)? Or the formal language of logic can handle this kind of writing?
Thanks.

REPLY [5 votes]: The proper usage of a formal notation or of a more informal one depends particularly on the context of presentation. It is essential to whom we communicate an idea and this should guide us to use a suitable level of formal notation.
Here is an excerpt from P.R. Halmos' instructive paper How to write Mathematics regarding the aspect:
What about the usage of logical symbols?

P. R. Halmos: Here is a sample :
"Prove that any complex number is the product of a non-negative number and a number of modulus $1$."
...
One way to recast the sample sentence of the preceding paragraph is to establish the convention that all "individual variables" range over the set of complex numbers and then write something like
  $$\forall z\exists p\exists u [(p=|p|) \wedge (|u|=1) \wedge (z=pu)]. $$
I recommend against it. The symbolism of formal logic is indispensable in the discussion of the logic of mathematics, but used as a means of transmitting ideas from one mortal to another it becomes a cumbersome code. The author had to code his thoughts in it (I deny that anybody thinks in terms of $\exists, \forall, \wedge$, and the like), and the reader has to decode what the author wrote ; both steps are a waste of time and an obstruction to understanding. Symbolic presentation, in the sense of either the modern logician or the classical epsilontist, is something that machines can write and few but machines can read.

Hint: You might also have a look at the related question Why there is no sign of logic symbols in mathematical texts?