# Biconditional (if-and-only-if)

Any two propositions **P** and **Q** can be joined with the biconditional operator, producing the new, complex, proposition:

**P if and only if Q**

The proposition **P if and only if Q** is true if and only if both **P** and **Q** are true, or if both **P** and **Q** are false. It is false only when one of them is true and the other false.

The biconditional is a complex operator, built out of simpler operators. Think of it this way:

**P if and only if Q**is the same as:**( If P then Q ) and ( P only if Q )**. This is like saying:**( If P then Q ) and ( If Q then P )**.

The **if and only if** operator plays a special role in definitions. When we say **P if and only if q**, we are saying that P *says the same thing* as Q.