"A paradox is a statement that contradicts itself..."
Actually, a statement that contradict itself is generally labeled 'false'. Example: "x is a natural number greater than 5 and less than 3". This pose no problem, and can be resolved by assuming that x is not in N, not greater than 5 or not less than 3, any combination of which makes the sentence false. With a typical 'logical paradox', any attempt of categorizing the sentence as false makes it true, which in turn makes it false etc.