The Contrapositive

A statement of the form “if p, then q” is known as a conditional statement. An example of such a statement is “If you are a cat, then you have fur.” Another common way to say this is “All cats have fur.”

There are 3 other statements that often come up in association with a conditional statement. Let’s use the example above to illustrate.

Conditional: If you are a cat, then you have fur.
Converse: If you have fur, then you are a cat.
Inverse: If you are not a cat, then you do not have fur.
Contrapositive: If you do not have fur, then you are not a cat.

The most important thing to know for standardized tests such as the SHSAT is that the contrapositive is logically equivalent to the original conditional statement! The converse and inverse are not.

For example, suppose the conditional statement “All cats have fur” is true. You may want to rewrite this as “If you are a cat, then you have fur.” It follows that “If you do not have fur, then you are not a cat” is also true.

In particular, if you are given the statement “Skittles does not have fur,” you can infer “Skittles is not a cat.”

Note that neither the converse nor the inverse is logically equivalent to the original conditional statement, but they are equivalent to each other.

Example 1

All of Jim’s friends can ski.

If the statement above is true, which of the following statements must also be true?

(A) If John cannot ski, then he is not Jim’s friend
(B) If Jeff can ski, then he is not Jim’s friend.
(C) If Joseph can ski, then he is Jim’s friend.
(D) If James is Jim’s friend, then he cannot ski.
(E) If Jordan is not Jim’s friend, then he cannot ski.

Solution: The given statement can be written in conditional form as “If you are Jim’s friend, then you can ski.” The contrapositive of this statement is “If you cannot ski, then you are not Jim’s friend. Replacing “you” with “John” gives the correct answer as choice (A).

Example 2

If a beverage is listed in menu A, it is also listed in menu B.

If the statement above is true, which of the following statements must also be true?

(A) If a beverage is listed in menu B, it is also in menu A.
(B) If a beverage is not listed in menu A, it is not listed in menu B.
(C) If a beverage is not listed in menu B, it is not listed in menu A.
(D) If a beverage is not listed in menu B, it is in menu A.
(E) If a beverage is listed in menu B, it is not listed in menu A.

Solution: Simply observe that the statement in choice (C) is the contrapositive of the given statement. So the answer is choice (C).

Conclusion

If you understand how to identify the contrapositive of a conditional statement, then you will be able to easily get most logic questions correct that appear on standardized tests.

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