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

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.

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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

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

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Well, you surely can't say the first is the same expression as the second. They are equivalent expressions. –  Peter Smith Jan 9 '13 at 22:45
+1; exactly what I was coming in here to say - canonically, I would consider this to not be a WFF by most rules. –  Steven Stadnicki Jan 9 '13 at 22:49
@Peter: Again that depends on how you define syntax. Is the choice of glyph part of the identity of the expression, or are the glyphs just used to indicate repeated appearances of the same variable, or some other similar sort of thing? If the glyph is part of the identity of an expression, then indeed they would not be identical expressions. I was trying to avoid the word "equivalent" to avoid confusion with $\leftrightarrow$. –  Hurkyl Jan 9 '13 at 22:49