Here’s one of the most famous of all paradoxes:
A prisoner is told that he will be hanged on some day between Monday and Friday, but that he will not know on which day the hanging will occur before it happens. He cannot be hanged on Friday, because if he were still alive on Thursday, he would know that the hanging will occur on Friday, but he has been told he will not know the day of his hanging in advance. He cannot be hanged Thursday for the same reason, and the same argument shows that he cannot be hanged on any other day. Nevertheless, the executioner unexpectedly arrives on some day other than Friday, surprising the prisoner.
There are various arguments resolving the paradox (with no consensus), including one that claims the paradox still applies when there are only two days.
I have what may be a slight variation. With a problem this popular, it’s hard to be original, so I doubt it, but —
Let’s suppose it’s Friday, and the prisoner is told they will be hung today. They’re also told it will be unexpected which day they will be hung.
The prisoner detects an immediate contradiction, and so concludes he won’t be hung. He has an unpleasant surprise when he’s hung anyway.
The essential part here seems to be the “paradox” condition of the prisoner’s reasoning, which works like an extra day. In other words, approaching the problem with only the values “true” or “false” is not enough.