Multiple Sufficient and Necessary Conditions
Be careful with the contrapositive
To best understand this section you should be familiar with basic conditional reasoning. Many questions on the LSAT rely on the use of sufficient and necessary conditions, and a solid knowledge of this form of reasoning is essential to a strong test performance. This writeup is taken from the Lesson Two Homework of our full-length course, and our course homework assignments contain a number of writeups similar to this one.
Many statements on the LSAT contain either multiple sufficient conditions or multiple necessary conditions. Consider the following statement:
To graduate from Throckmorton College you must be both smart and resourceful.
In this statement there are two necessary conditions that must be satisfied if you are to graduate from Throckmorton: 1. you must be smart and 2. you must be resourceful. Thus, the proper diagram for this statement is:
| Graduate |  | Smart and Resourceful |
The difficulty in dealing with multiple necessary conditions comes with the contrapositive. In this case, if either one of the two necessary conditions is not met, then you cannot graduate from Throckmorton. It is not required that both necessary conditions fail to be met in order to stop the sufficient condition from occurring. Thus the proper diagram of the contrapositive of the statement is:
Note that in taking the contrapositive, the “and” in the necessary condition changed to “or.” The reverse would be true if the necessary conditions had originally been linked by the term “or.” Consider the following statement:
To graduate from Throckmorton College you must be smart or resourceful.
The proper diagram for this statement is:
| Graduate | | Smart or Resourceful |
In this case, to graduate from Throckmorton you need to meet just one of the two necessary conditions. Of course, this does not preclude the possibility of meeting both conditions, but it is not necessary for you to do so. Now let us take the contrapositive of the second statement:
This diagram indicates that if you are not smart and also not resourceful, then you cannot graduate from Throckmorton College. Again note that the “or” joining the necessary conditions has changed to “and.” Now let us examine a statement with two sufficient conditions. Consider the following statement:
If you are rich and famous, then you are happy.
The proper diagram for this statement is:
Rich and Famous | | Happy |
Please note that you must be both rich and famous to meet the sufficient condition, and that if you were both rich and famous then you would also be happy. Now let us take the contrapositive:
The contrapositive indicates that if you are not happy, then you are either not rich or not famous. Thus, if you are not happy, then you are not rich, not famous, or not both. Note that once again the “and” has become “or.” Consider the following statement:
If you are rich or famous, then you are happy.
The proper diagram for this statement is:
Rich or Famous | | Happy |
Note that you can be either rich or famous, or both, to meet the sufficient condition. Now let us take the contrapositive:
The contrapositive indicates that if you are not happy, then you are neither rich nor famous. Thus, if you are not happy, then you are not rich and not famous. Also note that once again the “or” has become “and.”
Quick Review
Remember, when taking the contrapositive, “and” becomes “or” and vice versa.
Statement I: If A, then B and C.
Diagram:
A | | B and C |
Contrapositive:
Statement II: If A, then B or C.
Diagram:
A | | B or C |
Contrapositive:
Statement III: If A or B, then C.
A or B | | C |
Contrapositive:
Statement IV: If A and B, then C.
Diagram:
A and B | | C |
Contrapositive: