Necessity and sufficiency

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 P then Q", Q is necessary for P, because the truth of Q is guaranteed by the truth of P (equivalently, it is impossible to have P without Q).[1][2] Similarly, P is sufficient for Q, because P being true always implies that Q is true, but P not being true does not always imply that Q is not true.[3]

In general, a necessary condition is one that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition.[4] 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.[5] That is, the two statements must be either simultaneously true, or simultaneously false.[6][7][8]

In ordinary English, "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. For example, being a male is a necessary condition for being a brother, but it is not sufficient—while being a male sibling is a necessary and sufficient condition for being a brother. Any conditional statement consists at least one sufficient condition and at least one necessary condition.