Use of the existential quantifier over multiple individuals

Question

I have a relatively simple logic problem. Basically, I want to state that there is an assignment of values $x,y,u,v,z$ that makes the statement $P(x,y,u,v,z)$ false.

Should I write $\exists x, y, u, v, z. \neg P(x,y,u,v,z)$ or $\exists x. \exists y. \exists u. \exists v. \exists z. \neg P(x,y,u,v,z)$, or something else?

Answer 1

More generally:

"$\exists x,y \in S\ ( P(x,y) )$" denotes "$\exists x \in S\ \exists y \in S\ ( P(x,y) )$", when $P$ is a binary predicate on $S$.

So what you have written are both conventionally understood as equivalent, and the former is easier to read. For an example of why it is preferable consider the following theorem: $ \def\nn{\mathbb{N}} \def\zz{\mathbb{Z}} $

$\forall a,b \in \zz\ ( \neg \exists d \in \nn\ ( d > 1 \land \exists k,m \in \zz\ ( a=dk \land b=dm ) ) \to \exists x,y \in \zz\ ( ax+by=1 ) )$.

It would be way harder to read if it was written out in full.

In logic you might also frequently encounter further abbreviation such as "$\forall x_1,\dots,x_n$" denoting "$\forall x_1 \cdots \forall x_n$", when $n$ is a given natural number. Again, this is just for readability.