Definition:Disjunction

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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 both $p$ and $q$ are true.


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


$p \lor q$ 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:

$\ds \bigvee_{i \mathop = 1}^n \ p_i = \begin{cases} p_1 & : n = 1 \\ & \\ \ds \paren {\bigvee_{i \mathop = 1}^{n - 1} \ p_i} \lor p_n & : n > 1 \end{cases}$

That is:

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


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

$\ds \bigvee P$

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


Truth Function

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

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


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:

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


Disjunct

The substatements $p$ and $q$ are known as the disjuncts.


Semantics of the Disjunction

The disjunction is used to symbolise any statement in natural language such that at least one of two substatements are held to be true.

Thus, $p \lor q$ can be interpreted as:

  • $p$ or $q$
  • $p$ or $q$ or both
  • $p$ unless $q$
  • Unless $p$, $q$.


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
$p\ \mathsf{OR} \ q$
q$ Used in various computer programming languages
$p + q$
$\operatorname A p q$ Łukasiewicz's Polish notation


Also known as

The disjunction is also known as the logical sum.


The symbol $\lor$ comes from the first letter of the Classical Latin vel, whose meaning is or.

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.


Examples

Arbitrary Example

The statement:

Either $2 + 2 = 4$ or $6$ is an odd number

is a disjunction whose truth value is true, because one of the disjuncts is true .

The statement:

Either $2 + 2 = 4$ or $6$ is not an odd number

is a disjunction whose truth value is true, because both of the disjuncts is true .

The statement:

Either $2 + 2 = 5$ or $6$ is an odd number

is a disjunction whose truth value is false, because neither of the disjuncts is true .


Waiver of Premiums

Premiums will be waived in the event of sickness or unemployment

is a disjunction whose disjuncts are:

Premiums will be waived in the event of sickness
Premiums will be waived in the event of unemployment

the unspoken understanding being that in the event of both sickness and unemployment, premiums will also be waived.


Also see

  • Results about disjunction can be found here.


Sources

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