There's a puzzle I enjoy that exposes the problem, I think.
Assume there's two chests in front of you. One has treasure inside, the other is trapped. In front of the one on the left, there's a sign that says "This treasure is trapped"; in front of the one on the right, there's a sign that says "Exactly one of these signs is truthful". Which chest do you open?
Solution:
If the sign on the right is truthful, then the one on the left has to be false, hence the treasure is on the left. But if the sign on the right is false,
then the one on the left still has to be false, so the treasure is on the left still. So you should open the chest on the left.
You follow the logic and decide which chest to open. Yet, just before you open either:
I come in and switch both signs with each other, so that the one that used to be on the left is now on the right, and viceversa.
What do you do now? Which chest do you open?
This shows that you were starting the problem with a wrong assumption, which is that all statements are either True or False, which logically leads to spurious results, as all wrong assumptions are wont to do.