GMAT - Contrapositive - Review

CONTRAPOSITIVE RULE OF LOGIC
While no training in formal logic is required to answer argument questions, they are essentially questions about logic. So some knowledge of formal logic will give you a definite advantage.

To begin, consider the seemingly innocuous connective "if..., then...." Its meaning has perplexed both the philosopher and the layman through the ages. The statement "if A, then B" means by definition "if A is true, then B must be true as well," and nothing more. For example, we know from experience that if it is raining, then it is cloudy. So if we see rain falling past the window, we can validly conclude that it is cloudy outside.

There are three statements that can be derived from the implication "if A, then B"; two are invalid, and one is valid.

From "if A, then B" you cannot conclude "if B, then A." For example, if it is cloudy, you cannot conclude that it is raining. From experience, this example is obviously true; it seems silly that anyone could commit such an error. However, when the implication is unfamiliar to us, this fallacy can be tempting.

Another, and not as obvious, fallacy derived from "if A, then B" is to conclude "if not A, then not B." Again, consider the weather example. If it is not raining, you cannot conclude that it is not cloudy--it may still be overcast. This fallacy is popular with students.

Finally, there is one statement that is logically equivalent to "if A, then B." Namely, "if not B, then not A." This is called the contrapositive, and it is very important.

If there is a key to performing well on the logical reasoning section, it is the contrapositive.

To show the contrapositive's validity, we once again appeal to our weather example. If it is not cloudy, then from experience we know that it cannot possibly be raining.

We now know two things about the implication "if A, then B":

1) If A is true, then B must be true.
2) If B is false, then A must be false.

If you assume no more that these two facts about an implication, then you will not fall for the fallacies that trap many students.