So there's a few different ways the paradoxy paradox of paradoxes in today's comic might work.
Let's assume that I only ever say one thing: "Everything I say is a lie." The statement "everything I say is a lie" in isolation is a paradox because it implies that "everything I say is a lie" is also a lie. If that statement is a lie, then it is not the case that everything I say is a lie. If it is not the case that everything I say is a lie, then I am indeed lying when I say "everything I say is a lie," because the only way to lie about "everything I say is a lie" is to say something that is not a lie. The statement implies both that I always lie and that I do not lie about always lying. Its falseness proves its truth, and its truth proves its falseness, and so it is a paradox.
If what I just said is a paradox, then I tell the truth by saying "what I just said is a paradox." If I tell the truth then not everything I say is a lie. Since I have told the truth, the statement "everything I say is a lie" is no longer a paradox, because it suggests that I always lie, but does not imply that I am telling the truth about always lying (I have told the truth at least once, so I do not always lie. I am simply lying about always lying now.) Finally, since "everything I say is a lie" is no longer a paradox, I am lying when I say "what I just said is a paradox" and so everything I say is a lie again.
I'm not actually sure if this is totally correct and I'll have to think about it some more but MAN AM I EVER GLAD THAT THE ENTIRE HUMAN RACE HASN'T BEEN WIPED OUT BY METALLIC MONSTERS OF OUR OWN CREATION.