# Definition:Disjunction

## Definition

Disjunction is a binary connective written symbolically as $p \lor q$ whose behaviour is as follows:

$p \lor q$

is defined as:

Either $p$ is true or $q$ is true or possibly both.

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

The statements $p$ and $q$ are known as:

the disjuncts
the members of the disjunction.

$p \lor q$ is voiced:

$p$ or $q$

### General Definition

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

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

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

That is:

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

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

$\displaystyle \bigvee P$

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

### Truth Function

The disjunction connective defines the truth function $f^\lor$ as follows:

 $\displaystyle f^\lor \left({F, F}\right)$ $=$ $\displaystyle F$ $\quad$ $\quad$ $\displaystyle f^\lor \left({F, T}\right)$ $=$ $\displaystyle T$ $\quad$ $\quad$ $\displaystyle f^\lor \left({T, F}\right)$ $=$ $\displaystyle T$ $\quad$ $\quad$ $\displaystyle f^\lor \left({T, T}\right)$ $=$ $\displaystyle T$ $\quad$ $\quad$

### Truth Table

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

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

### Boolean Interpretation

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

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

## Notational Variants

Various symbols are encountered that denote the concept of disjunction:

Symbol Origin Known as
$p \lor q$ 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica vee or vel
$p\ \mathsf{OR} \ q$
$p + q$
$\operatorname A p q$ Łukasiewicz's Polish notation

## Also known as

The disjunction is also known as the logical sum.

The disjuncts are thence known as the summands of the logical sum.

The symbol $\lor$ comes from the first letter of the Classical Latin vel.

This usage of or, that allows the case where both disjuncts are true, is called inclusive or, or the inclusive disjunction. In natural language the term and/or is often seen, especially in the case of legal documents.

Some sources refer to this as the weak or, where the strong or is used in the sense of the exclusive or.

$p \lor q$ is also called the logical alternation, or just alternation, of $p$ and $q$.

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