Where to put the "such that", given multiple quantifier

Question

Personally, I would put the "such that" (i.e. the symbol $:$ or $|$) behind any quantification. That is given an assertion $A(x,y)$, I'd write $$ \forall x\in X\exists y\in Y:A(x,y)\\ \exists x\in X\forall y\in Y:A(x,y) $$ However, the lecturer giving the first graders course I'm tutoring writes $$ \exists x\in X:\forall y\in Y A(x,y) $$ in the second case (and the same as me in the first case), wich looks "wrong" to me. However, googling didn't produce any source that talks about placement of "such that" at all, so the question is: is there any rule or guideline where to put the "such that", and if so, what is it?

Answer 1

If you are talking at the colon (:), it is just a convenient delimiter for readability. You could also have written equivalently:

$\exists x\in X:\forall y\in Y : A(x,y)$.

Or you could leave them out entirely:

$\exists x\in X\space \forall y\in Y\space A(x,y)$.

Answer 2

'Such that' typically introduces a restriction or condition that in many statements we do not want.

If we say 'for all $x$ in $S$, property $P$ holds', we are saying something different than if we say, 'for all $x$ in $S$ such that property $P$ holds,...'.

In the former case we are saying that $P$ holds for all $x\in S$. In the latter case we are saying we are looking at those $x\in S$ for which $P$ holds (which may or may not be all $x$ in $S$.

Answer 3

The "such that" is not required by the syntax of first-order logic.

The official syntax is :

$\forall x \exists y (x < y)$

which is an example of : $\forall x \exists y A(x,y)$.

The "such that" (i.e. the symbol ":" or "|") is used with the set builder notation :

$\{ x : \varphi(x) \}$ or $\{ x | \varphi(x) \}$

meaning : "the set of objects in the domain of discourse such that $\varphi$ holds of".

---

But the issue is with the "good" translation into natural language of the above formulae.

With :

$∀x∃yA(x,y)$

the natural reading is :

"for all $x$, there is an $y$ such that ..."

while for

$∃x∀yA(x,y)$

it "sounds better" :

"there is an $x$ such that, for all $y$ ...".