This sentence is false. Is that sentence true or false? If it's true, it's false. If it's false, it's true. This is the liar's paradox, and philosophers have been debating it for thousands of years. From what I can tell, it's tricky, but it can be resolved with a little inspection. We don't need to throw out classical logic.
What happens when an irresistible force meets an immovable object?
Uhhh.... if an "irresistible" force exists, then BY DEFINITION, an immovable object cannot exist. And vice versa.