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 $$\left({p \implies q}\right) \and \left({q \implies p}\right)$$

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:
 * $${\left({p \implies q}\right) \quad \left({q \implies p}\right) \over p \iff q} \qquad \qquad {p \iff q \over p \implies q} \qquad \qquad {p \iff q \over q \implies p}$$

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 $$\left({p \implies q}\right) \and \left({q \implies p}\right)$$

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.