Mistaking the Necessary and Sufficient Conditions

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A question on the Top Law Schools message board caught our eye this week:

Is there a difference between ‘mistaking the sufficient condition for the necessary condition’ and ‘mistaking the necessary condition for the sufficient condition’? I can sort of see a difference, but I feel like it could be phrased either way and still be the same flaw.

This is a great question.  As it turns out, the two things have slightly different meanings.  If you mistake the sufficient condition for the necessary condition, you treat the thing that really is the necessary condition as if it were the sufficient condition.  If you mistake the necessary condition for the sufficient condition, you treat the thing that really is the necessary condition as if it were the sufficient condition.

Ultimately the two mistakes amount to the same thing, though, because by making one, you automatically make the other as well.  The result: the dreaded Fallacy of the Converse.

This is all fairly complicated, so let’s see an example.  A craigslist job posting for a Program Manager position includes the following:

Must have above average knowledge of Microsoft Office Suite.

There’s a conditional statement implied here.  Having Microsoft skills is a condition for getting the job.  But what kind of condition – necessary or sufficient?

If you read the rest of the job listing, it’s clear that having Microsoft skills is necessary but not sufficient.  It’s required, but it’s not the only thing that’s required.  Candidates also must have strong analytical ability, excellent communication skills, an understanding of process and problem solving, etc.  Not everyone who has Microsoft skills will get the job, but anyone who gets the job will have Microsoft skills.

So, this conditional can be diagrammed as:

Get the Job –> Microsoft Skills

If you mistake Microsoft skills for a sufficient condition, you’d think that having these skills would be enough to guarantee you’d get hired.  Your diagram would look like:

Microsoft Skills –> Get the Job

If you didn’t end up getting the job, you’d feel wronged:

“Sorry, we went with another candidate.”
“But I have above average knowledge of Microsoft Office Suite!”
“Yeah, but you’re a French Lit major and you were drunk during your interview.”
“Merde!”

This would be a case of mistaking a necessary condition (Microsoft skills) for a sufficient condition.  Let’s now see a case in which you might mistake a sufficient condition for a necessary condition.

If you’re the president, you’ll be famous.

Here, being president is a sufficient condition for being famous.  It is enough to guarantee fame. If we mistook it for the necessary condition, we’d understand something rather different. In that case, being president would be the only route to fame, and every famous person would have to be a president.

We are liable to make both mistakes.  To treat conditions that really are sufficient as necessary, and conditions that really are necessary as sufficient.

But upon closer inspection, the two mistakes amount to the same thing, because one entails the other.  Take a look back at our diagrams for Program Manager posting:

RIGHT: Get the Job –> Microsoft Skills
WRONG: Microsoft Skills –> Get the Job

We’ve mistaken a necessary condition (Microsoft skills) as a sufficient condition.  But we’ve also mistaken a sufficient condition (getting the job) for a necessary one.

When we originally looked at this conditional, we understood that Microsoft skills were a necessary condition for getting the job.  But we could have just as rightly said that getting the job is a sufficient condition for having Microsoft skills.  If Hannah got hired, that would guarantee that she knew Excel – no one without those skills would even be considered.

When we confused one, we confused both. Same deal with our famous president example.  By mistaking “being president” for a necessary condition, we also mistook fame for a sufficient condition, as if being famous were enough to guarantee you were president.

What all of these confusions boil down to is the Fallacy of the Converse.  This is when we erroneously “flip” the two sides of a conditional.  We make what should be the sufficient condition the necessary, and we treat what should be the necessary as the sufficient.  The resulting conditional will be the converse of what you want, and both of the conditions will be misplaced.

The “converse” of A –> B is B –> A.  They look similar but they don’t mean the same thing!

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