Necessary and Sufficient Conditions

necessary condition is something that is required for something else to happen or to hold true. For example, in order for somebody to be a licensed medical doctor they must have attended university. Necessary conditions, however, are not always sufficient for an outcome to occur – they are needed, but may not be enough by themselves to guarantee the outcome. In the case of our previous example, having attended university is necessary in order for somebody to be a licensed medical doctor, however it is not sufficient, for one may have attended university to study something totally unrelated to medicine. A sufficient condition is one that is enough all by itself to guarantee the outcome. In the case of the doctor, a sufficient condition might be be having passed a certain qualifying exam, since everyone who passes this exam by the very fact of passing becomes a licensed medical doctor.

Necessary and sufficient conditions are often confused. To avoid such confusion, one must understand not only the difference between them, but also the four possible sets of relationships between the two concepts: something could be neither necessary nor sufficient, it could be necessary but not sufficient, it could be sufficient but not necessary, or it could be both necessary and sufficient. To illustrate these four possibilities, consider the situation of determining whether somebody has the right to vote in an upcoming election:

  • Neither necessary nor sufficient: that somebody has black hair is neither necessary nor sufficient to have the right to vote. They don’t need to have black hair (e.g. brown hair is also acceptable), so it is not necessary, nor does having black hair guarantee that one will have a right to vote (e.g. one could be too young).
  • Necessary but not sufficient: that somebody be over the age of 18 is a necessary condition for having the right to vote, for any people below that age are not permitted to vote. However it is not a sufficient condition, since some people over the age of 18 still do not have the right to vote (e.g. certain convicted criminals).
  • Sufficient but not necessary: if somebody was permitted to vote in the previous election and they have not since become ineligible, then they will be permitted to vote in this election. This is a sufficient condition, since anyone satisfying this condition will always have the right to vote. However, it is not a necessary condition, because there are some people who do not meet this condition who nevertheless still are eligible to vote (e.g. those who were too young at the last election but are now over 18).
  • Necessary and sufficient: that somebody is over 18 years of age and is a citizen with no disqualifying characteristics (e.g. a criminal record) is both necessary and sufficient to be eligible to vote. Possessing these characteristics is sufficient for eligibility, since anyone who has them automatically has a right to vote. These characteristics are also necessary for being eligible, since everyone who can vote must meet all these conditions. They are thus both necessary and sufficient.

Further Reading

Necessary versus sufficient conditions: Brief definitions with some examples

The concepts of necessary and sufficient conditions: Comprehensive definitions with numerous examples

Necessary and sufficient conditions: A more advanced discussion of the uses and problems with the concepts of necessity and sufficiency