This article discusses only the reasonable relationships implied by necessary and sufficient. For the causal'' meanings see causation.

In logic, the words necessary and sufficient describe problems that consist between propositions or states of affairs, if one is incidental 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.

• A necessary condition is one that must be satisfied for the result to happen. Breathing is necessary to stay alive; if you did not breathe, you would not stay alive. Breathing is not sufficient to stay alive, for if you did nothing but breathe, you could still die.

• A sufficient condition is one that, if it is satisfied, the result is certain to happen. Jumping is sufficient to leave the ground, since the act of jumping causes you to leave the ground. Jumping is not necessary to leave the ground however, since one could step onto a ladder and leave the ground in a way which isn't jumping.

• Sometimes conditions can be both necessary and sufficient. For example, "Being the Fourth of July " is both necessary AND sufficient for "being the United States Independence Day".

"Necessary and sufficient" is another way of saying the logical statement "if and only if" (sometimes abbreviated to "iff").

Necessary conditions

To say that P is necessary for Q is to say that "if P is not true, then Q is not true". 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" or "Q $\backslash Rightarrow$ P" (Q implies P), and may also be seen as "P, if Q", "P whenever Q" or "P when Q".

Example 1: Consider the statement "Being a rectangle is necessary for being a square." Here if you are not a rectangle then it is impossible for you to be a square. That is, if you are a square, then you are also automatically a rectangle.

Example 2: Suppose that any lightning bolt causes thunder (however quiet the thunder may be) and suppose that by "thunder" we mean the sound caused by lightning (not any other loud rumbling). Then it might be said "thunder is necessary for lightning", for if there is absolutely no thunder, then there cannot be any lightning. That is, if lightning does occur, then it must create some thunder.

Example 3: As an example of something NOT being a necessary condition, consider the rectangle/square example. Notice that being a square is NOT a necessary condition for being a rectangle, since there are rectangles which are not squares.

Sufficient conditions

To say that P is sufficient for Q is to say that P being true forces Q to be true, or whenever P occurs, Q occurs. The logical relation is expressed as "If P then Q" or "P $\backslash Rightarrow$ Q", and may also be seen as "P implies Q."

Example 1: For simplicity, let us suppose everyone is biologically male or female, and that a "father" is a biological male who has fathered a child. Then "being a father is sufficient for being male".

Example 2: As in the previous section, let us define "thunder" as the sound that lightning creates. Then "thunder is sufficient for lightning." For if one hears thunder, then some lightning must have occured in order to create the thunder.

Example 3: As an example of a condition being NOT sufficient, consider the "male/father" example. Being male is NOT sufficient for being a father, since there are males which are not yet fathers.

Relationship between "Necessary" and "Sufficient"

The statement that "P is sufficent for Q" is the same as "Q is necessary for P", for both statements are the same as "P implies Q".

Example: Recall that "Being a rectangle is necessary for being a square". Also, "being a square is sufficient for being a rectangle."

Necessary and sufficient conditions

To say that P is necessary and sufficient for Q is to say two things:

1. P is necessary for Q (Q $\backslash rightarrow$ P)
2. P is sufficient for Q (P $\backslash rightarrow$ Q)

For example, if Alice always eats steak on Monday, but never on any other day, it can be said "being Monday is a necessary condition for Alice eating steak." This is so since Alice does not eat steak on days that are not Monday. Also, "being Monday is a sufficient condition for Alice eating steak." This is true since Alice always eats steak on Monday.

Consider the thunder/lightning example as outlined in previous sections. "Thunder is necessary for lightning", since absolutely no thunder means there isn't any lightning to create any noise. "Thunder is sufficient for lightning" since thunder (as we have narrowly defined it) must have originated from some lightning.

The relationship between being a square and being a rectangle is one which is NOT "necessary and sufficient" despite the ordering of the conditions "square" and "rectangle". "Being a rectangle is necessary for being a square", yet "being a rectangle is NOT sufficient for being a square". "Being a square is sufficient for being a rectangle", yet "being a square is NOT necessary for being a rectangle."

"P is necessary and sufficient for Q" expresses the same thing as "P if and only if Q" (P$\backslash Leftrightarrow$Q).

See also

• If and only if
• Causation

External links

• Stanford Encyclopedia of Philosophy: Necessary and Sufficient Conditions

Logic | Technical terminology | Mathematical terminology

Notwendige und hinreichende Bedingung | תנאי שקול | 充分必要条件