"If and Only If" Usage

Logical formulation of a biconditional statement.
the logical statement for a biconditional shows how it is formed from a conditional and its converse. C.K.Taylor

When reading about statistics and mathematics, one phrase that regularly shows up is “if and only if.” This phrase particularly appears within statements of mathematical theorems or proofs. We will see precisely what this statement means.

To understand “if and only if” we must first know what is meant by a conditional statement. A conditional statement is one that is formed from two other statements, which we will denote by P and Q. To form a conditional statement, we could say “If P then Q.”

The following are examples of this kind of statement:

  • If it is raining outside, then I take my umbrella with me on my walk.
  • If you study hard, then you will earn an A.
  • If n is divisible by 4, then n is divisible by 2.

Converse and Conditionals

Three other statements are related to any conditional statement. These are called the converse, inverse and the contrapositive. We form these statements by changing the order of P and Q from the original conditional and inserting the word “not” for the inverse and contrapositive.

We only need to consider the converse here. This statement is obtained from the original by saying, “If Q then P.” Suppose we start with the conditional “If it is raining outside, then I take my umbrella with me on my walk” The converse of this statement is: “If I take my umbrella with me on my walk, then it is raining outside.”

We only need to consider this example to realize that the original conditional is not logically the same as its converse. The confusion of these two statement forms is known as a converse error. One could take an umbrella on a walk even though it may not be raining outside.

For another example, we consider the conditional “If a number is divisible by 4 then it is divisible by 2.” This statement is clearly true. However, this statement’s converse “If a number is divisible by 2, then it is divisible by 4” is false. We only need to look at a number such as 6. Although 2 divides this number, 4 does not. While the original statement is true, its converse is not.

Biconditional

This brings us to a biconditional statement, which is also known as an if and only if statement. Certain conditional statements also have converses that are true. In this case, we may form what is known as a biconditional statement. A biconditional statement has the form:

”If P then Q, and if Q then P.”

Since this construction is somewhat awkward, especially when P and Q are their own logical statements, we simplify the statement of a biconditional by using the phrase “if and only if.” Rather than say ”if P then Q, and if Q then P” we instead say “P if and only if Q.” This construction eliminates some redundancy.

Statistics Example

For an example of the phrase “if and only if” that involves statistics, we need look no further than a fact concerning the sample standard deviation. The sample standard deviation of a data set is equal to zero if and only if all of the data values are identical.

We break this biconditional statement into a conditional and its converse. Then we see that this statement means both of the following:

  • If the standard deviation is zero, then all of the data values are identical.
  • If all of the data values are identical, then the standard deviation is equal to zero.

Proof of Biconditional

If we are attempting to prove a biconditional, then most of the time we end up splitting it. This makes our proof have two parts. One part we prove “if P then Q.” The other part of the proof we prove “if Q then P.”

Necessary and Sufficient Conditions

Biconditional statements are related to conditions that are both necessary and sufficient. Consider the statement “if today is Easter, then tomorrow is Monday.” Today being Easter is sufficient for tomorrow to be Easter, however, it is not necessary. Today could be any Sunday other than Easter, and tomorrow would still be Monday.

Abbreviation

The phrase “if and only if” is used commonly enough in mathematical writing that it has its own abbreviation. Sometimes the biconditional in the statement of the phrase “if and only if” is shortened to simply “iff.” Thus the statement “P if and only if Q” becomes “P iff Q.”