I'm a complete layman, so my technical terms might be misleading. Sorry for the many small questions, it's just that I don't know how to formulate my question right.
What is the deal with paradoxes like the liar's/Epimenides' paradox, the Barber's and Russell's paradox, Turing's proof for the Halting problem etc, in which "if the answer is no, then yes, and if yes then no."?
Why are they so meaningful and baffling? How are they revolutionizing? How can they be used as proofs or refutations for theories? Isn't that a problem in which the answer affects the answer itself? And a bit further, do they imply something non-dual, even something like a superposition?
For all I know they just seem to me like a catchy riddle. Such as: "answer yes or no, will your next answer be 'no'?"