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Beyoncé Logic—A Commentary by Ian Ayres ’86

The following commentary was posted on newyorktimes.com on October 13, 2009.

Beyoncé Logic
By Ian Ayres ’86

Lots of great responses to my Doonesbury puzzler.

http://freakonomics.blogs.nytimes.com/2009/10/13/beyonce-logic/

Implicitly, Alex was arguing, “If you are an independent, then you have a mind of your own.”

From which she concludes, “Conversely, if you are not an independent,” then you do not have a mind of your own.

Alex, I think, is making both a mistake in English usage and a mistake in logic.

Her mistake in usage is that she should have said “inversely” instead of “conversely.” The converse of “If p, then q” is “If q, then p.” But the last frame concerns an inverse: “If not p, then not q.” An interesting empirical study would look to see how often newspapers or academics misuse these adverbs (I’m sure I have).

Her mistake in logic is that neither an inverse nor a converse needs to have the same truth value as the original conditional statement. So even if we accept her original claim (“If you are an independent, then you have a mind of your own”) as true, we need not accept her conclusion that the inverse is also true.

Her two mistakes might be related. In the third frame, she suggests that the definition of independent is having a mind of your own. A converse of a true definition will also be true. (If a figure has four sides, then the figure is a quadrilateral). So by defining the word, maybe Alex is subtly trying to bolster her logic that the converse must be true too. But then she baits and switches to the inverse.

The take-home lesson is that we should be more careful in using “conversely” and “inversely” in our speech and in drawing conclusions from converses and inverses. A weakness in common usage is that no one ever says “contrapositively.” But a contrapositive is the only reframing of a conditional statement that is assured to have the same truth value.

“If p, then q” implies contrapositively “If not q, then not p.”

Playing around with inverses, converses, and contrapositives is one of the more bizarre pastimes of my family. We see a billboard for an adjacent apartment complex that says “If you lived here, you’d be home already” and immediately reframe it: “If you are not home already, you don’t live here.”

On a recent drive from Kansas City to Columbia, Missouri, we had an extended conversation on the logic behind Beyoncé’s song “Single Ladies.” Is it really true that “If you liked it, then you should have put a ring on it”? One way to test your answer is to ask whether the contrapositive is true: “If you shouldn’t have put a ring on it, then you didn’t like it.”

Of course, there are many possible meanings of “liked it,” but the consensus in my family is that neither the statement nor its contrapositive are true (because you might have “liked it” but learned that the other person was married). However, a majority of us think that the inverse of the song’s claim is true: If you did not like it, then you shouldn’t have put a ring on it. And we know that the contrapositive of the statement must also have the same truth value. So we must also believe “If you should’ve put a ring on it, then you liked it.” (The inverse and converse of an original statement are contrapositives of each other!)

Conditional claims strangely are at the center of Beyoncé’s craft. Consider the truth value of her claims in “If I were a boy…”