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:
“Old-fashioned determinism was what we may call hard determinism. It did not shrink from such words as fatality, bondage of the will, necessitation, and the like. Nowadays, we have a soft determinism which abhors harsh words, and, repudiating fatality, necessity, and even predetermination, says that its real name is freedom; for freedom is only necessity understood, and bondage to the highest is identical with true freedom.”
—William James (1842–1910)
“It is necessary to posit something which is necessary of itself, and has no cause of its necessity outside of itself but is the cause of necessity in other things. And all people call this thing ‘God.’”
—Thomas Aquinas (c. 1225–1274)