Definition:Conditional

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Definition

The conditional or implication is a binary connective:

$p \implies q$

defined as:

If $p$ is true, then $q$ is true.


This is known as a conditional statement.

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


$p \implies q$ is voiced:

if $p$ then $q$

or:

$p$ implies $q$


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

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


The rest of this article interprets the conditional in its usual mathematical sense.

That is, $p \implies q$ is considered to be true whenever:

$p$ is false

or:

$q$ is true.

This interpretation of implication is called logical implication or material implication.

Its most important property is that the truth of $p \implies q$ does not a priori establish a relation of causality between $p$ and $q$.

This allows for the conditional to be truth-functional, at the price of a slight mismatch with its use in natural language.


Truth Function

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

\(\displaystyle f^\to \left({F, F}\right)\) \(=\) \(\displaystyle T\)
\(\displaystyle f^\to \left({F, T}\right)\) \(=\) \(\displaystyle T\)
\(\displaystyle f^\to \left({T, F}\right)\) \(=\) \(\displaystyle F\)
\(\displaystyle f^\to \left({T, T}\right)\) \(=\) \(\displaystyle T\)


Truth Table

The characteristic truth table of the conditional (implication) operator $p \implies q$ is as follows:

$\begin{array}{|cc||c|} \hline p & q & p \implies q \\ \hline F & F & T \\ F & T & T \\ T & F & F \\ T & T & T \\ \hline \end{array}$


Boolean Interpretation

The truth value of $\mathbf A \implies \mathbf B$ under a boolean interpretation $v$ is given by:

$v \left({\mathbf A \implies \mathbf B}\right) = \begin{cases} T & : v \left({\mathbf A}\right) = F \text{ or } v \left({\mathbf B}\right) = T \\ F & : \text{otherwise} \end{cases}$


and the truth value of $\mathbf A \impliedby \mathbf B$ under a boolean interpretation $v$ is given by:

$v \left({\mathbf A \impliedby \mathbf B}\right) = \begin{cases} T & : v \left({\mathbf A}\right) = T \text{ or } v \left({\mathbf B}\right) = F \\ F & : \text{otherwise} \end{cases}$


Semantics of the Conditional

$p \implies q$ can be stated thus:

  • If $p$ is true then $q$ is true.
  • $q$ 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.

  • $p$ is true therefore $q$ is true.
  • $p$ is true entails that $q$ is true.
  • $q$ is true because $p$ is true.
  • $p$ may be true unless $q$ is false.
  • Given that $p$ is true, $q$ is true.
  • $q$ is true whenever $p$ is true.
  • $q$ is true provided that $p$ is true.
  • $q$ is true in case $p$ is true.
  • $q$ is true assuming that $p$ is true.
  • $q$ is true on the condition that $p$ is true.


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:

Weak

In a conditional $p \implies q$, the statement $q$ is weaker than $p$.


Strong

In a conditional $p \implies q$, the statement $p$ is stronger than $q$.


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


Superimplicant

In a conditional $p \implies q$, the statement $p$ is superimplicant to $q$.


Subimplicant

In a conditional $p \implies q$, the statement $q$ is subimplicant to $p$.


Antecedent

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


Consequent

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


Necessary Condition

Let $p \implies q$ be a conditional statement.

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.


Sufficient Condition

Let $p \implies q$ be a conditional statement.

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.


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

Let $p \implies q$ be a conditional statement.

Let its consequent $q$ be true.

Then it is a fallacy to assert that the antecedent $p$ is also necessarily true.

That is:

$p \implies q, q \not \vdash p$


Denying the Antecedent

Let $p \implies q$ be a conditional statement.

Let its antecedent $p$ be false.

Then it is a fallacy to assert that the consequent $q$ is also necessarily false.

That is:

$p \implies q, \neg p \not \vdash \neg q$


Formal Implication

Formal implication is a usage of an implication in which it is necessary for there to be a formal connection between the antecedent and the consequent in order for the implication to have any semantic meaning.


Further Definitions

Converse

The converse of the conditional:

$p \implies q$

is the statement:

$q \implies p$


Inverse

The inverse of the conditional:

$p \implies q$

is the statement:

$\neg p \implies \neg q$


Contrapositive

The contrapositive of the conditional:

$p \implies q$

is the statement:

$\neg q \implies \neg p$


Notational Variants

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

Symbol Origin Known as
$p \implies q$ Implies
$p \to q$ often used when space is limited
$p \supset q$ 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica hook or horseshoe
$p \, \mathop {-\!\!\!<} q$ Charles Sanders Peirce sign of illation
$\operatorname C p q$ Łukasiewicz's Polish notation


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 known as

A conditional statement is also known as a hypothetical, an implication or an implicative statement.


Also see

  • Results about the conditional operator can be found here.


Sources