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In Logic , the words ''necessary'' and ''sufficient'' describe problems that consist between propositions or states of affairs, if one is accidental on the other. A necessary and sufficient condition, then, is one which when working with others, must happen, ''and'' is all that needs to happen, for something else to be the case.
"Necessary and sufficient" is another way of saying the logical statement "'' If And Only If ''" (sometimes abbreviated to "''iff''"). NECESSARY CONDITIONS To say that Q is ''necessary'' for P is to say that P ''cannot'' be True unless Q is true, or that ''whenever'' (wherever, etc.) P is true, so is Q. We might say that being at least sixteen years old is necessary for having a 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'' P is, there is Q - ie, fire (P) cannot occur without there being smoke (Q). 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." {Link without Title} In either case, the important thing is to note that one thing is assumed (a license, fire), and a second thing is derived as "necessarily following." Being at least 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. So, you can be sixteen without having a driver's license, and there are ways to generate smoke without fire. If Q is a necessary condition for P, then the logical relation between them is expressed as "If P then Q" or "P Q" (P Implies Q), and may also be seen as "Q, if P" or "Q whenever P" or "Q when P". SUFFICIENT CONDITIONS To say that Q is ''sufficient'' for P is to say that Q being true forces P to be true, or whenever Q occurs, P occurs. That there is a fire is sufficient for there being smoke. Necessary and sufficient conditions are therefore related. P is a necessary condition for Q just in case Q is a sufficient condition for P. In the sense in which we are using the word "sufficient", we might also say "Having a license is sufficient for being at least sixteen." This is confusing, since having a license doesn't ''cause'' you to be at least sixteen; still, the ordinary sense of it is that ''if'' you have a license, you must be at least 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 sufficient condition for another thing (smoke, age) - sufficient in the sense of "enough for the other to be the case". Importantly, a sufficient condition, by definition, is what cannot occur ''without'' the thing it is a condition for. So, you cannot have a license without being at least sixteen. If Q is a sufficient condition for P, then the logical relation between them is expressed as "If Q, then P" or "Q P", and may also be seen as "Q implies P." NECESSARY AND SUFFICIENT CONDITIONS To say that P is ''necessary and sufficient'' for Q is to say two things: # P is necessary for Q. # P is sufficient for Q. For example, if Alice always eats steak on Monday, but never on any other day, we might say "Being Monday is a necessary and sufficient condition for Alice eating steak." The converse is also true: "Alice eating steak is a necessary and sufficient condition for it being Monday". Thus, whenever P is necessary and sufficient for Q, Q is necessary and sufficient for P. Once again, this is confusing, since Alice's act of eating steak doesn't cause it to be Monday. Since the phrase "necessary and sufficient" can express a relation between sentences ''or'' between states of affairs, objects, or events, it should not be too quickly conflated with logical equivalence. Alice eating steak is not logically equivalent to it being Monday. However, "P is necessary and sufficient for Q" does express the same thing as "P If And Only If Q" (PQ). FOOTNOTES # For the purposes of this example, we're ignoring the possibility of fire that ''doesn't'' create smoke. SEE ALSO EXTERNAL LINKS
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