The director of the school is either deaf or fool, if . . . . . . . .

The correct phrase to complete the sentence is not both.

The sentence implies that the director cannot be both deaf and a fool at the same time.

Option A: neither would imply that the director is neither deaf nor a fool, which changes the meaning of the sentence.

Option B: both not is grammatically incorrect.

Option D: nothing does not fit the context of the sentence.