I was watching this video on statements. There is an example:
$x + \frac12 = 2$
It's an open statement as the truth value could be
F depending on the value of $x$.
$∃x: x + \frac12 = 2$ and $x ∈ ℤ$
Now the statement is closed statement
(proposition) as the truth value is
But later the professor said:
$x + \frac12 = 2$ and $x ∈ ℤ$
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
Update from the author of the video:
$x$ 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.