If n is a perfect square, then n is not prime. I hope you'll agree that this is a true statement for any n . These expressions are flamboyant ways of asserting the antecedent to be false. But note how we do it. And this sarcasm exactly follows the logic of material implication.
False antecedents create true conditionals.
At least take comfort from this. But the perversity violates only our ordinary expectations, not any logical principles. For material implication, despite its perversity, suffices to capture the logically essential core of meaning in ordinary uses of implication. Application to arguments Conditional statements have corresponding arguments, just as arguments have corresponding conditionals. The answer is that they carry over only in a special sense, and not in a simple or straightforward way.