Definition:Conditional

Definition
The conditional is a binary connective written symbolically as $p \implies q$ whose behaviour is as follows:


 * $p \implies q$

is 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:

Truth Table
As $\implies$ is not commutative, it is instructive to give truth tables for both $p \implies q$ and $q \implies p$ (which of course is the same as $p \ \Longleftarrow \ q$).

The truth table of $p \implies q$ and its complement is as follows:


 * $\begin{array}{|cc||c|c|} \hline

p & q & p \implies q & \neg \left({p \implies q}\right) \\ \hline F & F & T & F\\ F & T & T & F\\ T & F & F & T\\ T & T & T & F\\ \hline \end{array}$

The truth table of $p \ \Longleftarrow \ q$ and its complement is as follows:


 * $\begin{array}{|cc||c|c|} \hline

p & q & p \ \Longleftarrow \ q & \neg \left({p \ \Longleftarrow \ q}\right) \\ \hline F & F & T & F \\ F & T & F & T \\ T & F & T & F \\ T & T & T & F \\ \hline \end{array}$

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:

Affirming the Consequent
If a conditional holds, and its consequent is true, it is a fallacy to assert that the antecedent is true. That is: $p \implies q, q \not \vdash p$.

Denying the Antecedent
If a conditional holds, and its antecedent is false, it is a fallacy to assert that the consequent is false. That is: $p \implies q, \neg p \not \vdash \neg q$.

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:

It is usual in the context of mathematics to use "$\implies$", 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.

Also see

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