Necessity and Sufficiency - Sufficiency

Sufficiency

To say that P is sufficient for Q is to say that, in and of itself, knowing P to be true is adequate grounds to conclude that Q is true. (It is to say, at the same time, that knowing P not to be true does not, in and of itself, provide adequate grounds to conclude that Q is not true, either.) The logical relation is expressed as "If P then Q" or "P Q," and may also be expressed as "P implies Q." Several sufficient conditions may, taken together, constitute a single necessary condition, as illustrated in example 5.

Example 1: Stating that "John is a bachelor" implies that John is male. So knowing that it is true that John is a bachelor is sufficient to know that he is a male.

Example 2: A number's being divisible by 4 is sufficient (but not necessary) for its being even, but being divisible by 2 is both sufficient and necessary.

Example 3: An occurrence of thunder is a sufficient condition for the occurrence of lightning in the sense that hearing thunder, and unambiguously recognizing it as such, justifies concluding that there has been a lightning bolt.

Example 4: A U.S. president's signing a bill that Congress passed is sufficient to make the bill law. Note that the case whereby the president did not sign the bill, e.g. through exercising a presidential veto, does not mean that the bill has not become law (it could still have become law through a congressional override).

Example 5: That the center of a playing card should be marked with a single large spade (♠) is sufficient for the card to be an ace. Three other sufficient conditions are that the center of the card be marked with a diamond (♦), heart (♥), or club (♣), respectively. None of these conditions is necessary to the card's being an ace, but their disjunction is, since no card can be an ace without fulfilling at least (in fact, exactly) one of the conditions.