It's only a paradox if they're both true. If one or both is false then it's not a paradox. This would be the same as if I said 1+1=2 and 2-1=2 clearly they both can't be correct and yet if you try and make them both correct you have a clear problem.
If both are false then it's not a paradox simply because they are both wrong. It's only if they are both true that it's a paradox. Just because something is false doesn't mean the opposite is true. "The next quote is a lie" if false it could mean the next quote is true or it could be the word "next" is wrong or "is" could really be "was". This isn't a either/or situation it's only when you make that assumption that you interpret it to be that type of situation.
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