LSAT Contrapositive Sample Questions
Overview
It is practically impossible to succeed on the LSAT without understanding conditionals, which are statements in the form "If X, then Y." Throughout the test, you will be asked to interpret these statements and draw valid conclusions based on them. Consider this statement:
If something is a bird, then it has wings.
According to this statement, it is always the case that birds have wings. For the purposes of the LSAT, it does not matter whether this statement is true. It matters a great deal, however, what can be logically concluded from this statement.
We use an arrow (→) to represent the logical relationship between the two parts of the conditional.
"If it is a bird, then it has wings" can be symbolized:
B → W
We can also translate this statement as "Birds require wings" or "In order for it to be true that something is a bird, it must also be true that that thing has wings." In other words, having wings is a necessary condition of being a bird.
So what happens if we find out that something does not have wings? Since having wings is a necessary condition of being a bird, if something does not have wings, we know that it cannot be a bird. We symbolize this as:
~W → ~B Here the tilde ("~") means "NOT."
This statement is called the contrapositive. In order to create the contrapositive, you must reverse and negate both terms in the statement.
The contrapositive has the same truth value as the original conditional statement. In other words, if the original statement is true, the contrapositive must also be true.
If it is true, for example, that all bananas are fruit (B → F), then it must also be true that if an object is not fruit, then it is not a banana (~F → ~B).
Understanding the contrapositive will give you a tremendous advantage in the Logic Games and Logical Reasoning sections. Check out the rest of our LSAT sample questions for more practice before the big test.
Question 1
In order for the United States to decrease its carbon emissions, the gas mileage of cars on American highways must be greatly increased. However, because only a few car companies will actually attempt to increase the gas efficiency of their cars, any significant increase of gas mileage of these cars depends on most of these companies being successful in their attempts, which is extremely unlikely in the American market due to the high financial costs of thorough research. One possible way to eliminate the effects of these factors is to have the government provide incentives to car companies to produce such cars and subsequently subsidize their price to consumers.
Which one of the following is most strongly supported by the information above?
Explanation
Statements:
- In order for the United States to decrease its carbon emissions, the gas mileage of cars on American highways must be greatly increased.
- However, because only a few car companies will actually attempt to increase the gas efficiency of their cars, any significant increase of gas mileage of these cars depends on most of these companies being successful in their attempts, which is extremely unlikely in the American market due to the high financial costs of thorough research.
- One possible way to eliminate the effects of these factors is to have the government provide incentives to car companies to produce such cars and subsequently subsidize their price to consumers.
Symbolized
United States:
- decreased carbon emissions (↓CE) → significant increase in gas mileage of cars on highways (↑GM)
- Contrapositive: ~ ↑GM &rarr ~ ↓CE
Car Companies:
- significant increase in gas mileage of cars on highways (↑GM) → successful in their attempts (S)
- Contrapositive: ~ S → ~ ↑GM
Since the sufficient condition in (2) is the necessary condition in (1), we can combine the statements: ↓CE → ↑GM → S, and per the contrapositive, ~ S → ~ ↑GM → ~ ↓CE.
Choice D: This statement paraphrases the deductions made using the contrapositives of the statements above. If car companies are not successful in increasing efficiency, then there will not be an increase in the mileage of cars on American highways, and the U.S. will not decrease its carbon emissions:
(~ S → ~ ↑GM → ~ ↓CE)
Choice D is correct.
Choice A: (3) introduces a possible way of reducing carbon emissions; however (3) does not state this way's chances of success, so stating that the "United States will not be able to decrease its carbon emissions" is unsupported.
Choice B: Extreme. (3) describes one possible way of decreasing carbon emissions in the United States. This cannot be construed as the "best" way.
Choice C: While (2) states that the high financial costs of thorough research make it extremely unlikely that car companies will be successful in their attempts to increase gas efficiency, this cannot be construed as "it is almost impossible" to eliminate such costs.
Choice E: Extreme. While (1) states that increased gas mileage of cars is necessary to decrease carbon emissions in the United States, it does not indicate that this is the "most realistic way" to decrease America's carbon emissions.
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