In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then , is necessary for , because the truth of is "necessarily" guaranteed by the truth of . (Equivalently, it is impossible to have without , or the falsity of ensures the falsity of .) Similarly, is sufficient for , because being true always or "sufficiently" implies that is true, but not being true does not always imply that is not true.

In general, a necessary condition is one (possibly one of several conditions) that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary and sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false.

In ordinary English (also natural language) "necessary" and "sufficient" often indicate relations between conditions or states of affairs, not statements. For example, being round is a necessary condition for being a circle, but is not sufficient since ovals and ellipses are round but not circles – while being a circle is a sufficient condition for being round. Any conditional statement consists of at least one sufficient condition and at least one necessary condition.

In data analytics, necessity and sufficiency can refer to different causal logics, where necessary condition analysis and qualitative comparative analysis can be used as analytical techniques for examining necessity and sufficiency of conditions for a particular outcome of interest.

Definitions

In the conditional statement, "if S, then N, the expression represented by S is called the antecedent, and the expression represented by N is called the consequent. This conditional statement may be written in several equivalent ways, such as "N if S, "S only if N, "S implies N, "N is implied by S, , and "N whenever S.

In the above situation of "N whenever S, N is said to be a necessary condition for S. In common language, this is equivalent to saying that if the conditional statement is a true statement, then the consequent N must be true—if S is to be true (see third column of "truth table" immediately below). In other words, the antecedent S cannot be true without N being true. For example, in order for someone to be called Socrates, it is necessary for that someone to be Named. Similarly, in order for human beings to live, it is necessary that they have air.

One can also say S is a sufficient condition for N (refer again to the third column of the truth table immediately below). If the conditional statement is true, then if S is true, N must be true; whereas if the conditional statement is true and N is true, then S may be true or be false. In common terms, "the truth of S guarantees the truth of N. This contrasts with the probabilistic theory of concepts which states that no defining feature is necessary or sufficient, rather that categories resemble a family tree structure.

Simultaneous necessity and sufficiency

To say that P is necessary and sufficient for Q is to say two things:

  1. that P is necessary for Q, <math>P \Leftarrow Q</math>, and that P is sufficient for Q, <math>P \Rightarrow Q</math>.
  2. equivalently, it may be understood to say that P and Q is necessary for the other, <math>P \Rightarrow Q \land Q \Rightarrow P</math>, which can also be stated as each is sufficient for or implies the other.

One may summarize any, and thus all, of these cases by the statement "P if and only if Q, which is denoted by <math>P \Leftrightarrow Q</math>, whereas cases tell us that <math>P \Leftrightarrow Q</math> is identical to <math>P \Rightarrow Q \land Q \Rightarrow P</math>.

For example, in graph theory a graph G is called bipartite if it is possible to assign to each of its vertices the color black or white in such a way that every edge of G has one endpoint of each color. And for any graph to be bipartite, it is a necessary and sufficient condition that it contain no odd-length cycles. Thus, discovering whether a graph has any odd cycles tells one whether it is bipartite and conversely. A philosopher might characterize this state of affairs thus: "Although the concepts of bipartiteness and absence of odd cycles differ in intension, they have identical extension.

In mathematics, theorems are often stated in the form "P is true if and only if Q is true". <!--(The following is irrelevant and not true.) Their proofs normally first prove sufficiency, e.g. <math>P \Rightarrow Q</math>. Secondly, the opposite is proven, <math>Q \Rightarrow P</math>

  1. either directly, assuming Q is true and demonstrating that the Q circle is located within P, or
  2. contrapositively, that is demonstrating that stepping outside circle of P, we fall out the Q: assuming not P, not Q results.

This proves that the circles for Q and P match on the Venn diagrams above.-->

Because, as explained in previous section, necessity of one for the other is equivalent to sufficiency of the other for the first one, e.g. <math>P \Leftarrow Q</math> is equivalent to <math>Q \Rightarrow P</math>, if P is necessary and sufficient for Q, then Q is necessary and sufficient for P. We can write <math>P \Leftrightarrow Q \equiv Q \Leftrightarrow P</math> and say that the statements "P is true if and only if Q, is true" and "Q is true if and only if P is true" are equivalent.

See also

References

  • Critical thinking web tutorial: Necessary and Sufficient Conditions
  • Simon Fraser University: Concepts with examples