# If the answer is “no” then “yes” and vice versa type of paradoxes. What are they?

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'?"

Thanks.

• This is not a complete answer, but those paradoxes you mentioned are paradoxes of self-reference. en.wikipedia.org/wiki/List_of_paradoxes#Self-reference – Deepak May 21 '15 at 11:40
• Essentially, Russell's paradox shows that we should take care in how we define sets (which a basic building block of mathematics). Naively, a set can be defined as the collection of all objects which satisfy some property, but this doesn't always give well defined objects: the set $R$ of all sets not contained in themselves is neither in or outside $R$ itself, which is in contradiction with the law of excluded middle. It is not baffling in and of itself, but it tells us that we must look for a more refined way to define sets. – A.P. May 21 '15 at 11:47
• The "common pattern" to Russell's Paradox and Turing's Halting problem is the proof method : proof by contradiction. We assume that a certain "fact" $F$ holds and we derive a contradiction; thus, we conclude that the said "fact" does not hold, i.e. we conclude with $\lnot F$. In the first case is the "naive" assumption that for any "condition" expressible in the language of set theory, a set exists of all and only those objects satisfying the condition. In Turing's case the "fact" is the assumption that the so-called Halting problem can be "addressed" by a "computational" procedure. – Mauro ALLEGRANZA May 21 '15 at 12:12
• I wrote about the significance of Russell's paradox here: math.stackexchange.com/questions/1086645/… – MJD May 21 '15 at 12:16