Definition:Conjunction

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Definition

Conjunction is a binary connective written symbolically as $p \land q$ whose behaviour is as follows:

$p \land q$

is defined as:

$p$ is true and $q$ is true.


This is called the conjunction of $p$ and $q$.


$p \land q$ is voiced:

$p$ and $q$.


General Definition

Let $p_1, p_2, \ldots, p_n$ be statements.


The conjunction of $p_1, p_2, \ldots, p_n$ is defined as:

$\ds \bigwedge_{i \mathop = 1}^n \ p_i = \begin {cases} p_1 & \text{if } n = 1 \\ & \\ \ds \paren {\bigwedge_{i \mathop = 1}^{n - 1} p_i} \land p_n & \text{if } n > 1 \end {cases}$

That is:

$\ds \bigwedge_{i \mathop = 1}^n \ p_i = p_1 \land p_2 \land \cdots \land p_{n - 1} \land p_n$


In terms of the set $P = \set {p_1, \ldots, p_n}$, this can also be rendered:

$\ds \bigwedge P$

and is referred to as the conjunction of $P$.


Truth Function

The conjunction connective defines the truth function $f^\land$ as follows:

\(\ds \map {f^\land} {\F, \F}\) \(=\) \(\ds \F\)
\(\ds \map {f^\land} {\F, \T}\) \(=\) \(\ds \F\)
\(\ds \map {f^\land} {\T, \F}\) \(=\) \(\ds \F\)
\(\ds \map {f^\land} {\T, \T}\) \(=\) \(\ds \T\)


Truth Table

The characteristic truth table of the logical conjunction operator $p \land q$ is as follows:

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


Boolean Interpretation

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

$\map v {\mathbf A \land \mathbf B} = \begin{cases} \T & \text{if } \map v {\mathbf A} = \map v {\mathbf B} = \T \\ \F & \text{otherwise} \end{cases}$


Conjunct

The statements $p$ and $q$ are known as the conjuncts.


Semantics of the Conjunction

The conjunction is used to symbolise any statement in natural language such that two substatements are held to be true simultaneously.

Thus it is also used to symbolise the concept of but as well as and.

Thus $p \land q$ can be also interpreted as:

  • $p$ and $q$
  • $p$ but $q$
  • $p$, yet $q$
  • $p$, although $q$
  • $p$; still, $q$
  • $p$; however, $q$
  • $p$; on the other hand $q$
  • $p$; moreover $q$
  • $p$; furthermore, $q$
  • $p$; nevertheless, $q$
  • Not only $p$ but also $q$
  • Despite $p$, $q$


Warning

Beware of the usage of and in natural language which has the following form:

He fell out of bed and broke his leg

which does not mean the same as:

He broke his leg and fell out of bed.

This use of and actually means and then, as it is implicit that the two occurrences are neither simultaneous nor independent, but that the second occurrence happens as a result of the first.


Notational Variants

Various symbols are encountered that denote the concept of logical conjunction:

Symbol Origin Known as
$p \land q$ wedge
$p \mathop {\mathsf {AND} } q$
$p \mathop . q$ 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica dot
$p \mathop \And q$ Ampersand
$p \mathop {\And \And} q$ Used in various computer programming languages
$\operatorname K p q$ Łukasiewicz's Polish notation


Also known as

The conjunction is also known as the logical product.

The conjuncts are thence known as the factors of the logical product.

Some sources refer to the conjunction merely as and.


Treatments which consider logical connectives as functions may refer to this operator as the conjunctive function.


Examples

Roses and Violets

Roses are red and violets are blue

is a conjunction whose conjuncts are:

Roses are red
Violets are blue.


London and Manchester

London is a city and Manchester is a city

is a conjunction whose conjuncts are:

London is a city
Manchester is a city.


Also see

  • Results about conjunction can be found here.


Sources

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