The basic principles of logic dictate that a statement cannot be both true and false at the same time... and that a statement must be either true or false... Lucy is lying, or she is crazy, or she is telling the truth. She can’t be some combination, and she can’t be none of those things... The laws of logic dictate that she must be one of them... Notice how practical common sense about the suposed impossibility of other worlds doesn’t come into the equation.
