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Suppose I write something like:

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

or even

$\forall x \forall y P(x)$

In either case, the variable $y$ is quantified but does not appear in my predicate statement. Is this syntactically valid, and if so, how should $\exists y$ and $\forall y$ be interpreted in this context?

My intuition is that the extraneous quantifier should simply be ignored.

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    In general (I mean according to the syntactical specifications of most textbook) it is syntactically correct. From the "semantical" point of view, you are right : $∀x∃yP(x)$ as well as $∀x∀yP(x)$ have the same meaning of $∀xP(x)$. – 2017-01-23

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The kind of thing you are talking about is called 'Null Quantification'.

Here are the relevant equivalence principles:

Null Quantification

With $x$ a variable, and $\varphi$ a formula that does not contain $x$ as a free variable, we have:

$\exists x \varphi \Leftrightarrow \varphi$

$\forall x \varphi \Leftrightarrow \varphi$

The equivalences hold because for any interpretation, the exact same objects that satisfy the left side satisfy the right side ... Exactly because the quantifier has no bearing on this.

To give one particular case of this to make this even more intuitive: suppose $\varphi = P$ for some statement $P$. Then $\exists x P$ is true iff there is some object in the domain such that $P$ is true ... which of course is true just when$P$ is true.

So, in your case, we have that $\exists y P(x) \Leftrightarrow P(x)$, and therefore $\forall x \exists y P(x) \Leftrightarrow \forall x P(x)$. Same for the other one.

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    "Then ∃xP is true iff there is some object in the domain such that P is true" - this works even if the domain of x is the null set? – 2017-01-23
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    @alexw Good question! No, it does not work for the null set. But most logics assume 'existential import', which says that the domain is always non-empty ... which makes not only sense practically (what's the point of reasoning over an empty domain), but is often necessary for some of the inference rules defined. E.g. That from $\forall x P(x)$ I can infer $P(a)$ using Universal Instantiation or $\forall Elim$ depends on this very assumption. – 2017-01-23