Necessity and sufficiency

In logic, necessity and sufficiency are implicational relationships between statements. The assertion that one 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, "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. Being a male sibling is a necessary and sufficient condition for being a brother. Fred's being a male sibling is necessary and sufficient for the truth of the statement that Fred is a brother.

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 many equivalent ways, for instance, "N if S", "S implies N", "S only if N", "N is implied by S", S ⇒ N or "N whenever S".

In the above situation, we also say that N is a necessary condition for S. In common language this is saying that if the conditional statement is a true statement, then the consequent N must be true if S has any chance of being true (see "truth table" immediately below). Phrased differently, the antecedent S can not 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.

We also say that S is a sufficient condition for N. Consider the truth table again. If the conditional statement is true, then if S is true, N must be true. In common terms, "S" guarantees N". Continuing the example, knowing that someone is called Socrates is sufficient to know that that someone has a Name.

A necessary and sufficient condition requires that both of the implications S ${\displaystyle \Rightarrow }$ N and N ${\displaystyle \Rightarrow }$ S (which can also be written as S ${\displaystyle \Leftarrow }$ N) hold. From the first of these we see that S is a sufficient condition for N, and from the second that S is a necessary condition for N. This is expressed as "S is necessary and sufficient for N ", "S if and only if N ", or S ${\displaystyle \Leftrightarrow }$ N.

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