Necessity and Sufficiency - Necessity

Necessity

The assertion that P is necessary for Q is colloquially equivalent to "Q cannot be true unless P is true," or "if P is false then Q is false." By contraposition, this is the same thing as "whenever Q is true, so is P". The logical relation between them is expressed as "If Q then P" and denoted "Q P" (Q implies P), and may also be expressed as any of "P, if Q"; "P whenever Q"; and "P when Q." One often finds, in mathematical prose for instance, several necessary conditions that, taken together, constitute a sufficient condition, as shown in Example 5.

Example 1: In order for it to be true that "John is a bachelor," it is necessary that it be also true that he is
  1. unmarried
  2. male
  3. adult
since to state "John is a bachelor" implies John has each of those three additional predicates.
Example 2: For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number that is both even and prime.
Example 3: Consider thunder, in the technical sense, the acoustic quality demonstrated by the shock wave that inevitably results from any lightning bolt in the atmosphere. It may fairly be said that thunder is necessary for lightning, since lightning cannot occur without thunder, too, occurring. That is, if lightning does occur, then there is thunder.
Example 4: Being at least 30 years old is necessary of serving in the U.S. Senate. If you are under 30 years old then it is impossible for you to be a senator. That is, if you are a senator, it follows that you are at least 30 years old.
Example 5: In algebra, in order for some set S together with an operation to form a group, it is necessary that be associative. It is also necessary that S include a special element e such that for every x in S it is the case that e x and x e both equal x. It is also necessary that for every x in S there exist a corresponding element x" such that both x x" and x" x equal the special element e. None of these three necessary conditions by itself is sufficient, but the conjunction of the three is.

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Famous quotes containing the word necessity:

    “I complacently accepted the social order in which I was brought up. I probably would have continued in my complacency if the happy necessity of self-support had not fallen to my lot; if self-support had not deepened and widened my contacts and my experience.”
    —Mary Barnett Gilson (1877–?)

    “Private property is held sacred in all good governments, and particularly in our own. Yet shall the fear of invading it prevent a general from marching his army over a cornfield or burning a house which protects the enemy? A thousand other instances might be cited to show that laws must sometimes be silent when necessity speaks.”
    —Andrew Jackson (1767–1845)

    “There was a time when the average reader read a novel simply for the moral he could get out of it, and however naïve that may have been, it was a good deal less naïve than some of the limited objectives he has now. Today novels are considered to be entirely concerned with the social or economic or psychological forces that they will by necessity exhibit, or with those details of daily life that are for the good novelist only means to some deeper end.”
    —Flannery O’Connor (1925–1964)