Definition:Conditional

Definition
The conditional is a binary connective:
 * $p \implies q$

defined as:
 * If $p$ is true, then $q$ is true.

This is known as a conditional statement.

$p \implies q$ is voiced:
 * if $p$ then $q$

or:
 * $p$ implies $q$

We are at liberty to write this the other way round. $q \ \Longleftarrow \ p$ means the same as $p \implies q$.

$q \ \Longleftarrow \ p$ is sometimes known as a reverse implication.

Also known as
A conditional is also referred to as an implication. Both terms are used more or less interchangeably on.

It is also known as:
 * logical implication
 * material implication

but also see the Rule of Material Implication for a more specific usage of the latter term.

A conditional statement is also known as a conditional proposition or just a conditional.

uses the symbol $\supset$ for the conditional, instead of $\implies$. This is not done on.

Complement
The complement of $\implies$ does not have a recognised symbol of its own.

However, the complement of $p \implies q$ can of course be written $\neg \left({p \implies q}\right)$.

Truth Function
The conditional connective defines the truth function $f^\to$ as follows:

Fallacies Concerning the Conditional
If we know that $q$ is true, and that $p \implies q$, this tells us nothing about the truth value of $p$. This also takes some thinking about. Here is a plausible example which may illustrate this.

Let $P$ be the statement:
 * $x$ is a whole number divisible by $4$.

Let $Q$ be the statement:
 * $x$ is an even whole number.

It is straightforward to prove the implication $P \implies Q$. (We see that if $P$ is true, that is, that $x$ is a whole number divisible by $4$, then $x$ must be an even whole number, so $Q$ is true.) However, $Q$ can quite possibly be an even number that is not divisible by $4$, for example, $x = 6$. In this case, $Q$ is true, but $P$ is false.

To suppose otherwise is to commit a fallacy. So common are the fallacies that may be committed with regard to the conditional that they have been given names of their own:

Relationship between Inverse, Converse and Contrapositive
Notice that:
 * The inverse of a conditional is the converse of its contrapositive
 * The inverse of a conditional is the contrapositive of its converse
 * The converse of a conditional is the inverse of its contrapositive
 * The converse of a conditional is the contrapositive of its inverse.

Also see

 * Therefore
 * Because
 * Interderivable (Logical Equivalence)
 * Biconditional