Is $\vDash \exists x ( Q x \to \forall x Qx)$ a valid sentence?

Question

Is $\vDash \exists x ( Q x \to \forall x Qx)$ a valid sentence?

$Q$ is a unitary relation.

I suppose that $\vDash Q x \to \forall x Qx$ , which is equivalent to $\vDash Q x \to \forall y Qy$ is invalid, since there is a structure $\mathfrak{A}$ with the universe $|\mathfrak{A}|=\{a,b\}$ plus one relation $Q = \{a\}$ and a function $s$ which sends the variable $x$ to $a$. But I got confused henceforth this point.

I'm inclined to reason that, since $x$ is bounded, the part $\forall x$ is redundant, the sentence should be valid.

Answer 1

Whether it's even well-formed depends on the low-level details of how you define syntax.

But even if it is well-formed in the syntax you use, using a variable $x$ as a dummy variable in a context where $x$ already has meaning is usually a bad idea.

That said, typically in syntax that allows such a thing, a variable acquires the innermost meaning. Therefore

$$\exists x ( Q x \to \forall x Qx)$$

is the same expression as

$$\exists x ( Q x \to \forall y Qy)$$

and is a different expression than

$$\exists x ( Q x \to \forall y Qx)$$