From ProofWiki
Jump to navigation Jump to search


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:

$\displaystyle \bigwedge_{i \mathop = 1}^n \ p_i = \begin{cases} p_1 & : n = 1 \\ & \\ \displaystyle \left({\bigwedge_{i \mathop = 1}^{n-1} \ p_i}\right) \land p_n & : n > 1 \end{cases}$

That is:

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

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

$\displaystyle \bigwedge P$

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:

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


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 $p$, $q$

Notational Variants

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

Symbol Origin Known as
$p \land q$ wedge
$p\ \mathsf{AND} \ q$
$p \ . \ q$ 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica dot
$p \ \And \ q$ Ampersand
$\operatorname K p q$ 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.

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


Roses and Violets

Roses are red and violets are blue

is a conjunction whose conjuncts are:

Roses are red
Violets are blue.

Also see

  • Results about conjunction can be found here.