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:

Antecedent
In a conditional $p \implies q$, the statement $p$ is the antecedent.

Some authors use the term premise (or premiss), but we already have a use for the term premise.

Other authors use the term hypothesis, but this word has other applications (see hypothesis), so we prefer not to use it in this context.

Consequent
In a conditional $p \implies q$, the statement $q$ is the consequent.

Some authors use the term conclusion, but we already have a use for that term conclusion.

Sufficient Condition
If $p \implies q$, then $p$ is a sufficient condition for $q$.

That is, if $p \implies q$, then for $q$ to be true, it is sufficient to know that $p$ is true.

This is because of the fact that if you know that $p$ is true, you know enough to know also that $q$ is true.

Necessary Condition
If $p \implies q$, then $q$ is a necessary condition for $p$.

That is, if $p \implies q$, then it is necessary that $q$ be true for $p$ to be true.

This is because unless $q$ is true, $p$ can not be true.

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.

Comparison to Necessary and Sufficient Conditions in Modal Logic
The above discussion applies to propositional logic. Predicate logic can then be seen to build on the latter, while modal logic builds on the predicate.

You can see close similarity in the terms necessary and sufficient:


 * $\Box p$ (necessarily p) is equivalent to $\neg \Diamond \neg p $ ("not possible that not-p")
 * $\Diamond p $ (possibly p) is equivalent to $\neg \Box \neg p $ ("not necessarily not-p")

which are demonstrated by an application of De Morgan's Laws.

For a more in depth analysis than is currently available on ProofWiki, see the Wikipedia article on modal logic.