Basic Logic and the Contrapositive

December 14, 2010

Some standardized tests, such as the LSAT, contain verbal questions that rigorously test your knowledge of logic. The GMAT isn't so insistent, but some background in logic will still pay off.

Your logical skills most explicitly pay off in Critical Reasoning questions. In CR, it is helpful to have a working knowledge of some logic basics, including the concepts of the "converse," the "inverse," and the "contrapositive."

Here's a sample if-then statement:

If it is raining, then I carry an umbrella.

Logicians might abbreviate that, "if r then u."

If all you know about me is the information in that statement, consider what deductions you can make. What if it is not raining? Do you know whether I carry an umbrella? What if I am carrying an umbrella? Do you know whether it is raining?

That's where logic comes in.

The converse of the above statement is:

If I am carrying an umbrella, then it is raining.

In logic terms: "if u then r."

Is that a reasonable deduction? No! It might be true, but it might not. Converses are often tempting (after all, if I'm carrying an umbrella, it's more than likely that it is raining), but are not airtight.

Let's look at the inverse:

If it is not raining, then I do not carry an umbrella.

Or in the logic abbreviation: "if not r, then not u."

How about that--a reasonable deduction? Again, no! It's possible, but it cannot be logically inferred from the original claim. Our first statement only told us what happens when it is raining. Maybe I carry an umbrella all the time!

Finally, the contrapositive:

If I am not carrying an umbrella, then it is not raining.

In logic terms: "if not u, then not r."

This one is a valid deduction. Consider the other possibility. If I am not carrying an umbrella and it is raining, then we know from the initial statement that I am carrying an umbrella. That's a contradiction.

The contrapositive of any true if-then statement is also true. The converse and inverse might be true, but without additional information, it is not a logical inference.

Next time you hear an if-then statement ("If I get out of work before 6 tonight, I'll go to the gym straight from the office.") think about what deductions are valid. As with any underlying skill, the more you internalize the knowledge, the more you'll be able to rely on it when you take the GMAT.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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