In mathematics, necessity and sufficiency are more brief terms for the classical logic connectives "if", "only if", and (when combined) "if and only if" (or "iff", if you prefer). Despite their brevity, they are quite correct and intuitive.

Necessity and sufficiency are best explained through examples. For these examples, I assume little knowledge of mathematics or logic.

• Let m and n be integers (i.e. whole numbers). Let 3m = n. Then n is necessarily a multiple of three. This is equivalent to saying "if m and n are both integers, and n is exactly triple whatever m is, then n is a multiple of three" in classical logic - or, even better, "n is an even number only if n is triple m". In symbol form, this is written (∃n∈J) 3m = n => n is a multiple of three (a quick glossary: "∃" is shorthand for "there exists at least one instance", "∈" means "in", J is the set of all integers, and "=>" is the "implication" symbol).
• Let m be an integer. To prove that m is an even number, it is sufficient to prove that m is divisible by 4. Again, in classical logic: "If m is divisible by 4, then it is even." Again, in symbolic form: (∃n∈J) m/4 = n => m is an even number.
• For a non-mathematical example: Let today be January 26th. This is a necessary and sufficient condition for proving that today is Australia Day. If it's January 26, then it is Australia day, and if it is Australia Day, it is January 26. Or, even better, "It is January 26 if and only if it is Australia Day". There's basically no two-ways about it. I won't provide a symbolic form here, as it can get messy, but the equivalent logical connective is "if and only if" and the symbol is <=>.

Note that in the middle example, I deliberately said "divisible by 4" instead of by 2. To prove that a number is even, you can divide it by four, six, eight, and so on. To divide it by 2 is sufficient and necessary. That is, amazingly, almost everything related to necessity and sufficiency (a little deeper understanding of mathematics or logic is required for more).

Now, you should be able to answer the question posed in Dodgeball: "Necessary?! Is it necessary for me to drink my own urine?"

The distinction between necessary and sufficient is important in logic and rhetoric (and therefore, we would like to believe, in debate and politics), although it is often ignored by those who speak to the masses.

Something is necessary if X cannot be achieved without it. Something is sufficient if something can be achieved with X, but might also be achieved without X.

A classic reminder of the difference is "a sledgehammer is sufficient to drive a nail into a board, but it is not necessary".

Aside from the common rhetorical error of calling sufficient things necessary ("It is necessary to reduce government expenditures in order to reduce the national deficit"), other common fallacies include identifying one necessary and/or sufficient condition but ignoring others ("If you you want to cut taxes, vote for me" [... and reduce services]) and assuming that there is only one sufficient and/or necessary condition ("Our current gun control policy is not sufficient to reduce crime, therefore we must strengthen it"...[or find another way to lower crime rates]).

The ideas represented by necessary and sufficient are very frequently an important part of the issues presented in public debate, but because people find these terms vaguely confusing they are often replaced by simpler words -- "we need", "it works -- and sometimes dropped altogether. When listening to a logical argument you should be ready to rephrase the argument using these terms, and to question any argument that falls apart when rephrased with a more strict formulation.

Log in or register to write something here or to contact authors.