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Necessary and sufficient conditions

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Necessary and sufficient conditions is a logical phrase that describes the relationship between two statements when one statement is true if and only if another statement is true. If a given statement is a necessary condition of another it must be true for the second statement to be true. If a given statement is a sufficient condition for another statement to be true the first statement, if satisfied as true, assures the second statement is true.

In physical science terms, a necessary and sufficient condition is known as the "cause" of an event, and the ensuing event is the "effect" of the prior one. The cause, of course, precedes the effect in time, although under the "necessary and sufficient" constraint, either event's occurrence can be imputed from the occurrence of the other.

In mathematics, "necessary and sufficient" is used to establish an equivalence relation between two statements: if the truth of one statement is necessary and sufficient for the truth of another statement, then the two statements are equivalent.

A related concept is dispositivity; that the truth of one statement may be either necessary, sufficient or both for the falsehood of another.

[edit] Corollaries of the concepts

Suppose a statement D, whose truth may be conditional on the truth of three other statements A, B and C (for the sake of argument, consider the list exhaustive). Given the notions of necessity and sufficiency described above, a few rules become clear.

  • If exactly one of the statements A, B or C is a necessary condition, but not a sufficient one, for the truth of D, no single condition among them is sufficient for the truth of D.
  • If one or more of the conditions A, B or C is sufficient but not necessary, then no single condition among the three is necessary for the truth of D.
  • If one of the conditions is both necessary and sufficient by itself, sufficiency of the others is immaterial (see the physical science point above).

There is also terminology governing more complex relationships involving necessity and sufficiency:

  • If A and B are both necessary but neither one individually, nor the two together, is sufficient for D, A and B may be said to be severally necessary but jointly insufficient.
  • If A, B and C are all necessary and the three together are sufficient for the truth of D, they may be said to be both severally necessary and jointly sufficient.
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