Editing Necessity and sufficiency (section)

==Sufficiency==

[[File:ICE 3 Fahlenbach.jpg|thumb|right|200px|That a train runs on schedule is a sufficient condition for a traveller arriving on time (if one boards the train and it departs on time, then one will arrive on time); but it is not a necessary condition, since there are other ways to travel (if the train does not run to time, one could still arrive on time through other means of transport).]]

If ''P'' is sufficient for ''Q'', then knowing ''P'' to be true is adequate grounds to conclude that ''Q'' is true; however, knowing ''P'' to be false does not meet a minimal need to conclude that ''Q'' is false.

The logical relation is, as before, expressed as "if ''P'', then ''Q''" or "''P'' ⇒ ''Q''". This can also be expressed as "''P'' only if ''Q''", "''P'' implies ''Q''" or several other variants. It may be the case that several sufficient conditions, when taken together, constitute a single necessary condition (i.e., individually sufficient and jointly necessary), as illustrated in example 5.

;Example 1:"John is a king" implies that John is male. So knowing that John is a king is sufficient to knowing that he is a male.

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

;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:If the U.S. Congress passes a bill, the president's signing of the bill is sufficient to make it law. Note that the case whereby the president did not sign the bill, e.g. through exercising a presidential [[veto#United States|veto]], does not mean that the bill has not become a law (for example, it could still have become a law through a congressional [[veto override|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 single diamond (♦), heart (♥), or club (♣). 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 these conditions.