Definition:Biconditional

Definition
Material Equivalence is a binary connective written symbolically as $p \iff q$ whose behaviour is as follows:


 * $p \iff q$ is defined as $\left({p \implies q}\right) \land \left({q \implies p}\right)$

Thus, $p \iff q$ means:
 * $p$ is true if and only if $q$ is true


 * $p$ is (logically) equivalent to $q$


 * $p$ is true iff $q$ is true

$p \iff q$ can be voiced:
 * $p$ if and only if $q$.

Other names for this operator include:
 * Biconditional


 * Logical Equivalence


 * Logical Equality

It can be written:
 * $\displaystyle {\left({p \implies q}\right) \quad \left({q \implies p}\right) \over p \iff q} \qquad \qquad {p \iff q \over p \implies q} \qquad \qquad {p \iff q \over q \implies p}$

Boolean Interpretation
From the above, we see that the boolean interpretations for $\mathbf A \iff \mathbf B$ under the model $\mathcal M$ are:


 * $\left({\mathbf A \iff \mathbf B}\right)_{\mathcal M} = \begin{cases}

T & : \mathbf A_{\mathcal M} = \mathbf B_{\mathcal M} \\ F & : \text {otherwise} \end{cases}$

Complement
The complement of $\iff$ is the exclusive or operator.

Truth Function
The biconditional connective defines the truth function $f^\leftrightarrow$ as follows:

Truth Table
The truth table of $p \iff q$ and its complement is as follows:

$\begin{array}{|cc||c|c|} \hline p & q & p \iff q & p \oplus q \\ \hline F & F & T & F \\ F & T & F & T \\ T & F & F & T \\ T & T & T & F \\ \hline \end{array}$

Semantics of Equivalence
The concept of material equivalence has been defined as:


 * $p \iff q$ means $\left({p \implies q}\right) \land \left({q \implies p}\right)$

So $p \iff q$ means:
 * If $p$ is true then $q$ is true, and if $q$ is true then $p$ is true.

$p \iff q$ can be considered as a shorthand to replace the use of the longer and more unwieldy expression involving two conditionals and a conjunction.

If we refer to ways of expressing the conditional, we see that:


 * $q \implies p$ can be interpreted as $p$ is true if $q$ is true, and


 * $p \implies q$ can be interpreted as $p$ is true only if $q$ is true.

Thus we arrive at the usual way of reading $p \iff q$ which is: $p$ is true if and only if $q$ is true.

This can also be said as:


 * The truth value of $p$ is equivalent to the truth value of $q$.


 * $p$ is equivalent to $q$.


 * $p$ and $q$ are equivalent.


 * $p$ and $q$ are coimplicant.


 * $p$ and $q$ are materially equivalent.


 * $p$ is true exactly when $q$ is true.


 * $p$ is true iff $q$ is true. This is another convenient and useful (if informal) shorthand which is catching on in the mathematical community.

Necessary and Sufficient
If $p \iff q$, we can say that $p$ is necessary and sufficient for $q$.

This is a consequence of the definitions of necessary and sufficient conditions.

Notational Variants
Various symbols are encountered that denote the concept of material equivalence:

It is usual in mathematics to use $\iff$, as there are other uses for the other symbols.