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, conditional proposition or just a conditional.

It is also known as a (logical) implication.

$p \implies q$" 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.

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


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

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

... and the boolean interpretations for $\mathbf A \ \Longleftarrow \ \mathbf B$ under the model $\mathcal M$ are:


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

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

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:

Semantics of the Conditional
We have stated that $p \implies q$ means If $p$ is true, then $q$ is true.

Alternatively, it can be said as:


 * $q$ is true if $p$ is true.


 * (The truth of) $p$ implies (the truth of) $q$.


 * (The truth of) $q$ is implied by (the truth of) $p$.


 * $q$ follows from $p$.


 * $p$ is true only if $q$ is true.

The latter one may need some explanation. $p$ can be either true or false, as can $q$. But if $q$ is false, and $p \implies q$, then $p$ can not be true. Therefore, $p$ can be true only if $q$ is also true, which leads us to our assertion.


 * $p$ may be true unless $q$ is false.


 * $q$ is true whenever $p$ is true.


 * $q$ is true provided that $p$ is true.


 * $p$ is true therefore $q$ is true.


 * $q$ is true because $p$ is true.


 * $q$ is subalternate to $p$.


 * $q$ is subimplicant to $p$.

Weak and Strong
If $p \implies q$ then:
 * $p$ is stronger than $q$.


 * $q$ is weaker than $p$.

Thus we have the notion of certain theorems having a weak and a strong version.

The language of the conditional
The conditional has been discussed at great length throughout the ages, and a whole language has evolved around it. For now, here are a few definitions:

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.

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

In mathematics, as opposed to works concerned purely with logic, it is usual to use "$\implies$", as then it can be ensured that it is understood to mean exactly the same thing when we use it in the "mathematical" context. There are other uses in mathematics for the other symbols.

Also see

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