Definition:Conditional

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[edit] 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 $q$ ".

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.

[edit] 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:

$\left({\mathbf{A} \ \Longleftarrow \ \mathbf{B}}\right)_{\mathcal{M}} = \begin{cases} T & : \mathbf{A}_{\mathcal{M}} = T \text{ or } \mathbf{B}_{\mathcal{M}} = F \\ F & : \text {otherwise} \end{cases}$

[edit] 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 $\neg \left({p \implies q}\right)$ .

[edit] 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:

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

[edit] 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."
" $p$ is stronger than $q$ " (and, by the same coin, " $q$ is weaker than $p$ ").
" $q$ is subalternate to $p$ ."
" $q$ is subimplicant to $p$ ."

[edit] 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:

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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:

[edit] 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: $p \implies q, q \not \vdash p$ .

[edit] 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: $p \implies q, \lnot p \not \vdash \lnot q$ .

[edit] Further definitions

[edit] Converse

The converse of the conditional $p \implies q$ is the statement $q \implies p$ .

The converse of a true conditional is not necessarily true, and the converse of a false conditional is not necessarily false.

[edit] Inverse

The inverse of the statement $p \implies q$ is the statement $\lnot p \implies \lnot q$ .

The inverse of a true conditional is not necessarily true, and the inverse of a false conditional is not necessarily false.

[edit] Contrapositive

The contrapositive of the conditional $p \implies q$ is the statement $\lnot q \implies \lnot p$ .

The Rule of Transposition gives that a statement and its contrapositive have the same truth value.

[edit] 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.

[edit] Notational Variants

Alternative symbols that mean the same thing as $p \implies q$ are also encountered:

$p \to q$ ;
$p \supset q$ , in which usage " $\supset$ " is called the "hook" or "horseshoe" sign.

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 a more "mathematical" context. There are other uses in mathematics for the other symbols.