Biconditional Statement — Definition, Examples & How To Write

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

What is a biconditional statement?

A biconditional statement combines a conditional statement with its converse statement. Both the conditional and converse statements must be true to produce a biconditional statement.

If we remove the if-then part of a true conditional statement, combine the hypothesis and conclusion, and tuck in a phrase "if and only if," we can create biconditional statements.

Geometry and logic cross paths many ways. One example is a biconditional statement. To understand biconditional statements, we first need to review conditional and converse statements. Then we will see how these logic tools apply to geometry.

Conditional statements

In logic, concepts can be conditional, using an if-then statement:

1. If I have a pet goat, then my homework will be eaten.

2. If I have a triangle, then my polygon has only three sides.

3. If the polygon has only four sides, then the polygon is a quadrilateral.

4. If I eat lunch, then my mood will improve.

5. If I ask more questions in class, then I will understand the mathematics better.

6. If the quadrilateral has four congruent sides and angles, then the quadrilateral is a square.

Each of these conditional statements has a hypothesis ("If …") and a conclusion (" …, then …").

These statements can be true or false. Whether the conditional statement is true or false does not matter (well, it will eventually), so long as the second part (the conclusion) relates to, and is dependent on, the first part (the hypothesis).

Converse statements

To create a converse statement for a given conditional statement, switch the hypothesis and the conclusion. You may "clean up" the two parts for grammar without affecting the logic.

Take the first conditional statement from above:

• Hypothesis: If I have a pet goat …

• Conclusion: … then my homework will be eaten.

Create the converse statement:

• Hypothesis: If my homework is eaten …

• Conclusion: Then I have a pet goat.

• Converse: If my homework is eaten, then I have a pet goat.

This converse statement is not true, as you can conceive of something … or someone … else eating your homework: your dog, your little brother. Your homework being eaten does not automatically mean you have a goat.

Let's apply the same concept of switching conclusion and hypothesis to one of the conditional geometry statements:

• Conditional: If I have a triangle, then my polygon has only three sides.

• Converse: If my polygon has only three sides, then I have a triangle.

Converse statement examples

For, "If the polygon has only four sides, then the polygon is a quadrilateral," write the converse statement.

Try this one, too: "If the quadrilateral has four congruent sides and angles, then the quadrilateral is a square."

How to write a biconditional statement

The general form (for goats, geometry or lunch) is:

Because the statement is biconditional (conditional in both directions), we can also write it this way, which is the converse statement:

Notice we can create two biconditional statements. If conditional statements are one-way streets, biconditional statements are the two-way streets of logic.

• Conditional: If I have a triangle, then my polygon has only three sides. (true)

• Converse: If my polygon has only three sides, then I have a triangle. (true)

Since both statements are true, we can write two biconditional statements:

• I have a triangle if and only if my polygon has only three sides. (true)

• My polygon has only three sides if and only if I have a triangle. (true)

You can do this if and only if both conditional and converse statements have the same truth value. They could both be false and you could still write a true biconditional statement ("My pet goat draws polygons if and only if my pet goat buys art supplies online.").

Let's see how different truth values prevent logical biconditional statements, using our pet goat:

• Conditional: If I have a pet goat, then my homework will be eaten. (true)

• Converse: If my homework is eaten, then I have a pet goat. (not true)

We can attempt, but fail to write, logical biconditional statements, but they will not make sense:

• I have a pet goat if and only if my homework is eaten. (not true)

• My homework will be eaten if and only if I have a pet goat. (not true)

Biconditional statement symbols

You may recall that logic symbols can replace words in statements. So the conditional statement, "If I have a pet goat, then my homework gets eaten" can be replaced with a p for the hypothesis, a q for the conclusion, and a $\to$ for the connector:

For biconditional statements, we use a double arrow, $\Leftrightarrow$, since the truth works in both directions:

Biconditional statement examples

We still have several conditional geometry statements and their converses from above.

• Conditional: If the polygon has only four sides, then the polygon is a quadrilateral. (true)

• Converse: If the polygon is a quadrilateral, then the polygon has only four sides. (true)

• Conditional: If the quadrilateral has four congruent sides and angles, then the quadrilateral is a square. (true)

• Converse: If the quadrilateral is a square, then the quadrilateral has four congruent sides and angles. (true)

Try your hand at these first, then check below. The biconditional statements for these two sets would be:

1. The polygon has only four sides if and only if the polygon is a quadrilateral.

2. The polygon is a quadrilateral if and only if the polygon has only four sides.

3. The quadrilateral has four congruent sides and angles if and only if the quadrilateral is a square.

4. The quadrilateral is a square if and only if the quadrilateral has four congruent sides and angles.

More examples

See if you can write the converse and biconditional statements for these. You can "clean up" the words for grammar.

• If I eat lunch, then my mood will improve.

Try doing it before peeking below!

• If I eat lunch, then my mood will improve. (true)

• If my mood improves, then I will eat lunch. (true)

Biconditional statements:

• I will eat lunch if and only if my mood improves.

• My mood will improve if and only if I eat lunch.

And now the other leftover:

• If I ask more questions in class, then I will understand the mathematics better. (true)

• If I understand the mathematics better, then I will ask more questions in class. (false)

You cannot write a biconditional statement for this leftover; the truth values are not the same.