--- aliases: [if and only if, biconditional, biconditional statement] --- #logic ## Definition > [!tldr] Definition > A **biconditional statement** is a [[Propositions|proposition]] that has the form "$A$ if and only if $Bquot; where $A$ and $B$ are [[Propositions|propositions]]. > > The biconditional statement "$A$ if and only if $Bquot; is written in mathematical notation as $A \leftrightarrow B$. Notes: * We consider the double arrow in $P \leftrightarrow Q$ to be a [[Connectives|connective]] that joins the [[Propositions|propositions]] $P$ and $Q$. * The [[Truth tables|truth table]] for the biconditional statement $P \leftrightarrow Q$ is: | $P$ | $Q$ | $P \leftrightarrow Q$ | | -- | -- | ---- | | T | T | T | | T | F | F | | F | T | F | | F | F | T | - In other words, **biconditional statements are true whenever the two propositions have the same truth value**. - The statement $P \leftrightarrow Q$ is [[Logical equivalence|logically equivalent]] to the statement $(P \rightarrow Q) \wedge (Q \rightarrow P)$. That is, the biconditional statement is true if the [[Conditional statements|conditional statement]] $P \rightarrow Q$ and its [[Converse|converse]] are both true, and false otherwise.