# When is the contrapositive the answer?

Just received this question from one of my online students:

When should we consider the contrapositive of a statement to be the answer? Should we always try to spend the extra time to write conditional statements both ways? I have lost points and time because I have eliminated all 5 answers... where one of them is actually the contrapositive of the correct answer.

Short answer: You should always consider the contrapositive when looking at a conditional (if-->then) statement. That's because a statement and its contrapositive are actually the exact same thing. Ready to get geeky? You know I am.

(What a geek might look like. Check out "The Inbetweeners" if you want a laugh... available on Netflix.)

The student above says he is concerned about "spending the extra time," but considering the contrapositive won't take any time once he's good enough at it. The statement "If X then Y" inherently implies "If Not Y, then Not X." At first, you will have to write this down. Eventually, you might not even need to write it down any more because you'll feel it in your bones. You'll realize that "If X then Y" and "If Not Y then Not X" are actually the exact same statement. It's not two different statements; it's the exact same statement, presented in two different ways. They are logically identical.

The "contrapositive as an answer choice" seems to appear most frequently on Sufficient Assumption questions, although it could happen on any type of question. Here's a quick example, using my Geto Boys post from last week:

Argument:

real gangsta-ass niggas don't flex nuts 'cause real gangsta-ass niggas know they got em

Last week I said that if the testmakers were to ask a Sufficient Assumption question based on this argument (like "Which one of the following, if true, would allow the conclusion to be properly drawn?") the answer could be "Nobody who has nuts would ever flex them." (Nuts-->NOT Flex.) That's a perfect answer for a Sufficient Assumption question because if it's true, then the argument wins.

But an equally good answer--actually, the exact same answer, worded differently--would be "Anybody who does flex does not have nuts." (Flex-->NOT Nuts.) That's "the contrapositive," sure... but it's also the exact same statement, logically.

OK, that's a whole lot of nerdery right there. I hope it's helpful! Let me summarize by saying this: If I ever predict that the correct answer should be "If X then Y," I won't only look for "If X then Y" in the answer choices. I will always also be looking for "If Not Y then Not X," because I know that's another way of saying the exact same thing.