Definition:Biconditional

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

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

Thus, $$p \iff q$$ means: "$$p$$ is true if and only if $$q$$ is true", or "$$p$$ is equivalent to $$q$$."

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

Other names for this operator include:
 * Biconditional;
 * Logical Equality.

You may encounter different symbols which mean the same thing as $$p \iff q$$, for example:


 * $$p \leftrightarrow q$$;
 * $$p \equiv q$$.

It is usual to use "$$\iff$$", as then it can be ensured that it is understood to mean exactly the same thing when we use it in a more "mathematical" context. There are other uses in mathematics for the other symbols.

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

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

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.

Looking back at ways of expressing the conditional, we see that:


 * $$q \Longrightarrow p$$ can be interpreted as "$$p$$ is true if $$q$$ is true," and
 * $$p \Longrightarrow q$$ 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.