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Formal Logic Questions

Tom will go to the movies only when his wife is out of town. He'll go to a matinee alone, but will see a movie at night only if accompanied, by Cruise and Katie.

This simple stimulus is made up of a couple of formal logic statements, each fraught with its own implications. Be on the lookout for Logical Reasoning stimuli that contain sentences that can be boiled down to such hard and fast rules. When you come across examples of these, you can apply the following principles of formal logic to help you arrive at the correct answer.

The Contrapositive

For any if/then statement - or a statement that can be translated into if/then form - the contrapositive of the statement will result in an equally valid second statement.

The contrapositive can be formed by reversing and negating the terms of any if/then statement. The general model goes like this:

If X, then Y.

The contrapositive of this statement is;

If not Y, then not X.

The contrapositive of a valid if/then statement will always be valid itself. Let's illustrate this with a simple example. Consider the following strict formal statement:

If the building has vacancies, then the sales office will be open.

To form the contrapositive, reverse and negate the terms, like so If the sales office is NOT open, then the building does NOT have vacancies. This would be a valid inference based on the original statement. The contrapositive, while quite a fancy term, is nothing more than everyday common sense.

Now let's apply the contrapositive to the first sentence of the earlier example.

Here's the original:

Tom will go to the movies only when his wife is out of town.

This is a little trickier, because it’s not stated in the form of a true if/then statement. But we can translate this statement into an if/then statement without changing its original meaning.

If Tom goes to the movies, then his wife must be out of town.

If the statement above is true, which one of the following must be true the basis of it? Why, the contrapositive of it, of course:

If Tom's wife is not out of town, then Tom does not go to the movies,

Wrong answers often result from either forgetting to switch around the terms before negating them, or negating only one of the terms. For example, from the above example, if Tom doesn’t to the movies, we can't infer anything about whether his wife is in or out town. Similarly, if Tom's wife IS out of town, we can't tell for sure whether Tom goes to the movies or not. Students commonly make this mistake and it costs them signifcantly in Umat Section 1.

If one part of the formal logic statement contains a compound phrase then both parts of the phrase must be taken into account. For example let's take the other part of the stimulus above:

Tom will see a movie at night only if accompanied by Cruise and Katie.

Translation: If Tom sees a movie at night, then he's accompanied Cruise and Katie.

Contrapositive: If Tom is not with Cruise and Katie, then he does not see a movie at night.

Correct Interpretation: If either Cruise or Katie is missing, then Tom's out of luck. If he's with only one of them, or neither of them, then can't go to a night movie.

The contrapositive is an important rule of logic. It is essential that you learn what it is and how to use it.