Is there a difference between using two vs one universal quantifier for two variables?

Question

Is $\forall x \forall y$ equivalent to $\forall (x, y)$?

For example, here is the statement of a Symmetric Relation in both ways:

$\forall x \forall y[xRy \rightarrow yRx]$

$\forall (x,y)[xRy \rightarrow yRx]$

Do these have the same meaning. If so, is one preferred or more common? What about when you have 3 or more variables, it seems like repeating the $\forall$ would get unwieldy. So want to make sure there's a rule for applying universal quantifier on multiple variables.

Answer 1

$\forall (x,y)[xRy \rightarrow yRx]$

or

$\forall x,y \ [xRy \rightarrow yRx]$

is really just short-hand for:

$\forall x \forall y[xRy \rightarrow yRx]$

That is, you really are dealing with two universal quantifiers. There is no such thing as one quantifier for multiple variables.

This is because a statement like:

$\forall y[aRy \rightarrow yRa]$

is a logical consequence, and therefore should be derivable. If you treat $\forall x,y \ [xRy \rightarrow yRx]$ as having only one quantifier, then you couldn't really do this ... or at least you would need to make the quantifier elimination rules more complicated. Any formal semantics defined over the syntax would need to become more complicated as well.

Answer 2

With universal quantification (and likewise for existential quantification), the truth conditions of $\forall x \forall y \mathop. P(x,y)$ and $\forall x \mathop. \forall y \mathop. P(x, y)$ are the same.

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The first formula, $\forall x \forall y \mathop. P(x, y)$, means for all $x$ and $y$, it holds that $P(x,y)$. The second means for all $x$, it holds for all $y$ that $P(x, y)$. These sentences will be true or false in exactly the same situations.

Mathematical logic textbooks will usually directly define the meaning of $\forall x \mathop. \varphi$ where $x$ is exactly one variable, and define the multiple variable case as an orthographic abbreviation for repeated single quantification. In most settings, the two alternatives you describe are thus equivalent by definition.

However, if you take a different approach and directly formalize mutli-variable quantification, then you can prove that universally quantifying everything all at once and universally quantifying individually produces sentences with the same truth conditions as a theorem.

However, this equivalence would not hold for every quantifier. Take for example, the uniqueness quantifier $\exists!$. This quantifier $\exists! x \mathop. \varphi$ is usually treated as an orthographic abbreviation for $(\exists x. \varphi) \land (\forall u \mathop \forall v \mathop. \varphi(u) \land \varphi(v) \to u = v)$. I won't be adopting this convention here and will instead be treating uniqueness quantification as primitive.

$\exists! (x, y) \mathop. P(x, y)$ means that there exists a unique pair of entities $(x, y)$ such that $P(x, y)$ holds.

$\exists! x \exists! y \mathop. P(x, y)$ means that there exists a unique $x$ such that there exists a unique $y$ such that $P(x, y)$ holds.

These very clearly have different meanings.

For example, let the domain of discourse be the integers and let $P(x, y)$ hold if and only if $(x, y) = (0, 0)$ holds or $x = 2$ holds.

There is not a unique pair of $(x, y)$ such that $P(x, y)$ holds.

However, there is a unique $x$ such that there's a unique $y$ such that $P(x, y)$, namely that $x$ is zero.