ENCYCLOPEDIA 4U .com

 Web Encyclopedia4u.com

# Necessary and sufficient

Necessary and Sufficient expresses a relation of one state of affairs, or one sentence to another: One says "A is necessary and sufficient for B", or commonly, "A is a necessary and sufficient condition for B". The phrase can express a relation between sentences or between states of affairs, objects, or events. It should therefore not be too quickly conflated with iff or with logical equivalence.

To say that A is necessary and sufficient for B is to say two things: straightforwardly enough, (1) A is necessary for B, and (2) A is sufficient for B.

(1) To say that A is necessary for B is to say that B cannot occur without A occurring, or that whenever (wherever, etc.) B occurs, so does A. We might say that being sixteen is necessary for having your driver's license.

In the sense in which we are using the word "necessary" here, we might also say "Smoke is necessary for fire." This is confusing, since smoke comes after fire; but all we are saying is that wherever B is, there is A. We are trying not to say anything about the direction of time at all. Ordinary language would say "Smoke is a necessary outcome of fire."

In either case, the important thing is to note that one thing is assumed (fire, a license), and a second thing is derived as "necessary following." Being sixteen is the necessary condition in the first case; smoke is the necessary condition in the second (though, again, we ordinarily would not call it a "condition".) Importantly, it is quite possible for a necessary condition to occur on its own. We might say, a necessary condition can be forced to occur by whatever it is necessary for, but it cannot force that thing to occur.

If A is a necessary condition of B, then the logical relation between then is expressed as "If B then A" or "B-->A"

(2) To say that A is sufficient for B is to say precisely the converse: that A cannot occur without B, or whenever A occurs, B occurs. That there is a fire is sufficient for there being smoke.

In the sense in which we are using the word "sufficient", we might also say "Having a license is sufficient for being sixteen. This is confusing, since having a license doesn't cause you to be sixteen; still, the ordinary sense of it is that if you have a license, you must be sixteen (We consider licenses proof of age because we consider them sufficient for age in something like this sense). Try to ignore the causal relationship and the direction of time: we are looking at it just as a logical relationship.

In either case, note that one thing is assumed (fire, a license), and this same thing we are identifying as the sufficent condition for another thing (smoke, age)--sufficient in the sense of, "enough for the other to be the case." Most importantly, a sufficient condition, by definition, is what cannot occur without the thing it is a condition for, but the thing that it is a condition for may well be able to occur without it.

If A is a sufficient condition for B, then the logical relation between them is expressed as "If A then B" or "A-->B".

Generally, if A is necessary to B, then B is sufficient for A; likewise if A is sufficient for B then B is necessary to A. As a result, to say that A is necessary and sufficient for B is also to say that B is necessary and sufficient for A--that is, that they occur only together, and neither occurs without the other.

This is not precisely the case if one is using causal sense of necessity or sufficiency.

Content on this web site is provided for informational purposes only. We accept no responsibility for any loss, injury or inconvenience sustained by any person resulting from information published on this site. We encourage you to verify any critical information with the relevant authorities.