Definition:Conditional

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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:

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


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


Further definitions

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.


Inverse

The inverse of the conditional:

$p \implies q$

is the statement:

$\neg p \implies \neg q$

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


Contrapositive

The contrapositive of the conditional:

$p \implies q$

is the statement:

$\neg q \implies \neg p$

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


Relationship between Inverse, Converse and Contrapositive

Notice that:


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
$p \supset q$ Bertrand Russell and Alfred North Whitehead: Principia Mathematica (1910) hook or horseshoe
$\operatorname C p q$ Łukasiewicz's Polish notation


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


Sources

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