Think about it this way, the statement can't be true and it can't be false. If it's true, then it's false, if it's false then it's
true. Either way, you end up contradicting yourself. But it can't be neither true nor false, because that implies contradiction too.

"This statement is false" is the basic form of well known and
popularized antinomies. The statement is not necessarily either true or false: in any system of logic that allows "this statement is false", there is no consistent way of assigning "true" or "false" to all statements. Compare this to the similar statement "This statement is provable", which is indeed
provable.

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Nubian