A sentence doesn't have to be either true or false. "This statement is false" is a good example. Another one is that famous "Colorless green ideas sleep furiously."
Yes. If only it weren't so easy.
Yes. If only it weren't so difficult.
When you start looking into meaning, intention, and truth, you run up against all sorts of folks who've been pondering these things a whole lot longer than us.
Of course a sentence does not have to be true or false (in the wholly logical sense). But, then again: all's I been saying is that logic has bugger all to do with grammar. Let's not lose sight of the OP. That's why I mentioned "situational semantics". And, also why, I brought fiction into things. You can dismiss these things all you want, but it's like not voting in an election: once you've decided not to vote, you don't get to kvetch. At least not without more meaning.
OP
