Learn the 4 Things You Must Know Before Taking the GMAT!   Download it Now!

GMAT Critical Reasoning: Avoid the “Inversion” fallacy of logical reasoning

I was driving down the street the other day and passed a Sara Lee truck with delicious-looking pictures of pastries on the back and the following motto emblazoned in big, bold letters on the side:

“Nobody doesn’t like Sara Lee.”

My first thought was, yep, that sounds about right. I mean, who doesn’t love Sara Lee’s coffee cakes and brownie bites?

But then I thought, wait a minute, why did they express the sentiment as a double-negative? Why didn’t they just say, “Everybody likes Sara Lee”?

Isn’t that saying the same thing?

No, it’s not.

But that’s a common fallacy to assume it is, and it’s something to be on the lookout for on GMAT Critical Reasoning questions as well. Let’s take a closer look at an aspect of logical reasoning called contraposition.

## Logic 101: Contraposition

In logic, contraposition refers to the idea that a conditional statement (“if X, then Y”) is logically equivalent to its contrapositive. That is to say, if you negate and invert the antecedent and consequent of the original conditional statement, it will be logically equivalent to that original statement.

Okay, so that sounds like a lot of gibberish, and you definitely don’t need to memorize that definition for the GMAT.

But you do need to understand what it means. Let’s break it down.

Consider the following logical assertion: “All dogs have tails.”

That can be restated as the conditional “If something is a dog, then it has a tail.”

To use logical notation, we could write it as D (dog) —> T (tail).

The contrapositive, then, reverses the D and T and negates them. In other words, the contrapositive of D —> T is thus -T —> -D. In lay terms, that argument would be expressed, “If something doesn’t have a tail, it is not a dog.”

And that makes sense, right?

Saying “If something is a dog, then it has a tail” is logically equivalent to “If something doesn’t have a tail, then it’s not a dog.”

So that’s pretty straightforward.

But beware. There are two common fallacies that people make when it comes to conditional statements like that, and you’ll want to make sure you can recognize them on GMAT Critical reasoning questions:

Fallacy #1: Inversion. Inversion would say that -D —> -T. “If something isn’t a dog, then it doesn’t have a tail.” But that’s not logically equivalent to D —> T. Certainly a cat isn’t a dog, but it does have a tail. So does a horse. And a bunny rabbit. Etc. There are lots of ways to disprove the -D —> -T assertion, if D —> T is true. Thus an inversion is not logically equivalent to its original conditional argument.

Fallacy #2: Conversion. Conversion would say that T —> D. “If something has a tail, then it is a dog.” That’s obviously not the same as D —> T, either. Again, a cat has a tail, but it’s not a dog. Likewise with a horse. Etc. So here we run into the same problem as with the Inversion fallacy.

Now let’s go back and look at the Sara Lee Motto, “Nobody doesn’t like Sara Lee.” Assuming that’s a true statement, my suggested revision to “Everybody likes Sara Lee” would have been a simple inversion, which we just identified as Fallacy #1. Just because there isn’t anybody who doesn’t like Sara Lee, that doesn’t mean that everybody does.

And actually, I got proof of that when I asked my wife, out of curiosity: Do you like Sara Lee? Her response, and I kid you not, was: “I don’t dislike them.” In other words, she doesn’t not like Sara Lee. But that doesn’t mean that she likes them, either. She’s just sort of neutral toward them.

So I guess Sara Lee picked a good — and accurate — way of expressing their motto, after all!

## Application Example

With this little bit of Logic 101 under our belts, let’s see how this plays out in a real GMAT critical reasoning question, as follows:

Left-handed people suffer more frequently than do right-handed people from certain immune disorders, such as allergies. Left-handers tend to have an advantage over the right-handed majority, however, on tasks controlled by the right hemisphere of the brain, and mathematical reasoning is strongly under the influence of the right hemisphere in most people.

If the information above is true, it best supports which of the following hypotheses?

A. Most people who suffer from allergies or other such immune disorders are left-handed rather than right-handed.
B. Most left-handed mathematicians suffer from some kind of allergy.
C. There are proportionally more left-handers among people whose ability to reason mathematically is above average than there are among people with poor mathematical reasoning ability.
D. If a left-handed person suffers from an allergy, that person will probably be good at mathematics.
E. There are proportionally more people who suffer from immune disorders such as allergies than there are people who are left-handed or people whose mathematical reasoning ability is unusually good.

The question is asking us to draw a conclusion from the information given. That’s what a hypothesis is. To answer this question, then, we need to get crystal-clear on what the premises of the original argument are saying. Let’s be sure to write them on our scratch paper in a way that lends itself to the kind of analysis we did above.

Premise #1, as expressed in the first sentence, says that left-handed people are more likely to develop immune disorders like allergies. We can re-word that as an if-then statement: “If someone is left-handed, then they are more likely to develop immune disorders like allergies.” Using our short-hand from above, we can write it like this on our scratch paper:

Left-handed –> Allergies

Premise #2 says that left-handed people tend to be better at right hemisphere brain activities such as math. Reworded as an if-then statement, that might read: “If someone is left-handed, then they are more likely to be better at right hemisphere brain activities such as math.” Using short-hand, we can write premise #2 like this on our scratch paper:

Left-handed –> Math (right hemisphere activities)

At this point we’ve established our two logical if-then statements. Any hypothesis we draw from these two statements, then, must be logically sound and follow the rules of logic established above. We cannot, for example, hypothesize the “inversion” or “conversion” of either of those two premises.

Which brings us to the answer choices.

What do you notice about answer choice A? That’s the fallacy of “conversion,” isn’t it? It reverses premise #1, and essentially says that “If someone has allergies, then they’re likely left-handed.” Allergies –> Left-handed. But we know that’s not logically equivalent to Left-handed –> Allergies, and thus we can’t draw that conclusion. Answer choice A is therefore incorrect.

See how that works?

Continuing through the answer choices, answer choice B has a couple of problems. First, there’s nothing to suggest that the two premises are related. In other words, premise 1 relates left-handers to immune disorders. Premise 2 relates left-handers to right-brain functionality. But we have no idea how immune disorders and right-brain functionality relate, just that left-handers are more susceptible to each. Moreover, even if we could surmise that perhaps the two would mix since they have left-handers in common, we certainly can’t make the leap to “most” left-handed mathematicians suffering from allergies. The superlative word “most” renders answer choice B incorrect.

Answer choice D makes the same error of mixing the two premises. Premise #1 makes a claim about how left-handers relate to allergies. There will be some overlap, where a certain percentage of left-handers do in fact have allergies. But that’s separate and apart from premise #2. There’s nothing to suggest how immune disorders and right-brain functionality might be related. Just because a left-hander might in fact have allergies, we don’t know anything about how that might also relate to math ability. So answer choice D is incorrect.

In choosing between answer choice C and E, then, answer choice C is better because it doesn’t mix the two premises like answer choice E does. And in fact, it makes perfectly logical sense based on Premise #2 that if we’re looking at the world of high-mathematically-reasoning individuals, we’d expect there to be a disproportionate number of left-handers since we know left-handers tend to be controlled more by right-brain activities, such as math. Conversely, in a group of people with poor mathematical reasoning ability, we wouldn’t expect as many left-handers. Answer choice C is therefore correct.

Well done if you got that!

What did you like most about this lesson? Please leave your questions and comments below! And for more GMAT critical reasoning tips, strategies, and practice problems, check out our comprehensive GMAT Critical Reasoning video course HERE. Good luck!