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
- unmarried
- male
- 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.
Read more about this topic: Necessity And Sufficiency
Famous quotes containing the word necessity:
“The use of force alone is but temporary. It may subdue for a moment; but it does not remove the necessity of subduing again: and a nation is not governed, which is perpetually to be conquered.”
—Edmund Burke (1729–1797)
“Central heating, French rubber goods, and cookbooks are three amazing proofs of man’s ingenuity in transforming necessity into art, and of these, cookbooks are perhaps most lastingly delightful.”
—M.F.K. Fisher (b. 1908)
“To expect to increase prices and then to maintain them at a higher level by means of a plan which must of necessity increase production while decreasing consumption is to fly in the face of an economic law as well established as any law of nature.”
—Calvin Coolidge (1872–1933)