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] 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.[2]

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.[3] 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.[4][5][6]

In ordinary English (also natural language) "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 of at least one sufficient condition and at least one necessary condition.


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