What is a good example of a paradox?

[Edwardo Piet]

All lies are false statements, not all false statements are lies.

IMO, it's not even true that all lies are false statements. There's merely usually strong correlation (people who tell lies are usually making false statements), and that's all.

Take the following example that I have just thought up that I call Gary and the Nonexistent Unusual Rabbit:

Imagine a man, Gary, who has the delusion that a man-eating rabbit with two heads lives in his shed, and he keeps his shed locked for that reason (he doesn't want to be eaten by a two-headed man-eating rabbit), but he's actually more afraid of the doctors thinking he's crazy than he is of the two-headed man-eating rabbit because he also believes the rabbit's arms aren't very strong and the rabbit definitely won't be capable of breaking out of a locked shed.

Next, imagine that he confesses his delusion to a trusted friend, Bob, and Bob snitches on him to the doctors because Bob is understandably worried about Gary's sanity. When the doctors arrive, Gary yells to them "There is no two-headed man-eating rabbit in my shed!" out of fear of them taking him away to an acute mental health ward.

Well, in that case he really does believe that the two-headed man-eating rabbit really does exist, so he's actually lying to the doctors because he really does believe the very opposite to what he tells them. But this is all in spite of the fact that the very statement that he made, i.e.: "There is no two-headed man eating rabbit in my shed" is a completely true statement because there really is no two-headed man-eating rabbit in his shed at all.