Is a paradox a truth value?

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Most of my reading has lead me to the conclusion that, all statements are either true or false, but they cannot be both or neither. It's just sort of an on/off thing. If so, how do paradoxes exist. The liar paradox takes a statement, "this statement is false," and just turns it into a big mess. Wouldn't that mean there are three possible truth values: true, false, and a sort of null value?