More formally: A condition A is said to be necessary for a condition B, if (and only if) the falsity (/nonexistence /non-occurrence) [as the case may be] of A guarantees (or brings about) the falsity (/nonexistence /non-occurrence) of B.
If X is a necessary cause of Y, then the presence of Y necessarily implies the presence of X. The presence of X, however, does not imply that Y will occur.
In logic, the terms necessity and sufficiency refer to the implicational relationships between statements or propositions.
For a statment to be true, any necessary condition of that statement must be satisfied. Formally, a statement P is a necessary condition of a statement Q if Q implies P.
The logical relation is expressed as "If Q then P" is denoted "Q P" (Q implies P).
Modal logic deals with alethic modality concepts (possibily true and necessarily true):
A proposition is said to be:
▪ possible if it is not necessarily false (regardless of whether it actually is true or false);
▪ necessary if it is not possibly false;
▪ contingent if it is not necessarily false yet not necessarily true either (a limited case of possibility).
Because anything is possible that is not necessarily false, almost anything is possible and almost nothing is logically impossible. Thus, possibility is a very weak condition in that much that is logically possible is not necessarily true. In a real sense, logical possibility signifies very little because it inheres so much while explaining so litttle.
Disjunctions of sufficient conditions may achieve necessity, while conjunctions of necessary conditions may achieve sufficiency.
A sufficient condition is that condition which guarantees the occurence of an event.
More formally: A condition C is said to be sufficient for a condition D, if (and only if) the truth (/existence /occurrence) [as the case may be] of C guarantees (or brings about) the truth (/existence /occurrence) of D.
If S is a sufficient cause of T, then the presence of S necessarily implies the presence of T. However, another cause U may alternatively cause T. Thus the presence of T does not imply the presence of S.
A statement is true if a sufficient condition for its truth is satisfied. Formally, a statement P is a sufficient condition of a statement Q if P implies Q. The logical relation is expressed as "If P then Q" or "P Q," and may also be expressed as "P implies Q."
A condition that is both necessary and sufficient can produce the effect when acting alone.
The assertion that one statement is a necessary and sufficient condition of another indicates that one statement is true if and only if (iff) the latter is true. That is, either both statements are true, or both statements are false. The logical relation can be alternately expressed as "P is sufficient for Q" or as "Q is necessary for P", since both statements mean that P implies Q.
Brennan, Andrew, "Necessary and Sufficient Conditions", The Stanford Encyclopedia of Philosophy (Fall 2003 Edition), Edward N. Zalta (ed.), url.
Necessary and sufficient conditions (wikipedia)