at first I was quite confused because it seemed like zmjezhd was saying that because some sentence can't be verified as true, it therefore isn't grammatical.

That is the opposite of what I was trying to say. Most linguists separate semantics and logic from grammar. For this group of people there can be perfectly grammatical sentences, e.g., the example sin the OP. Though they be meaningless for semantic or logical reasons. Again, I apologize for the confusion.

Ceci n'est pas un seing.