I was watching this [video](https://youtu.be/u8VE1_njuUo?list=PLcjqUUQt__ZGLhwUacPm7_RKs2eJNFwco&t=1301) on statements. There is an example:

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> $x + \frac12 = 2$
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It's an **open statement** as the truth value could be `T` or `F` depending on the value of $x$.

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> $∃x: x + \frac12 = 2$ and $x ∈ ℤ$
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Now the statement is **closed statement ~~(proposition)~~** as the truth value is `F`.

But later the professor said:

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> $x + \frac12 = 2$ and $x ∈ ℤ$
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is an **open statement**. But it's not clear to me how it's an **open statement** as for any value of $x$ the statement is `F`?

**Update from the author of the video:**

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> $x$ is a free (unquantified) variable here, so by definition this is an open statement. But you've identified something that's a source of confusion. Many online sites say that a statement is open if its truth value depends on what values the variable(s) take on. That usually coincides with the definition I indicated above. But this example points out that the two definitions are not the same. While it's easy to slip into the "is the truth value known" definition (as I seemed briefly to do at around 19:40), the definition of an open statement that works best is that it's a statement with one or more free variables, as I point out in several places in this video.
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