For instance, the English language sentence "it is sunny or it is warm" can be represented in logic using the disjunctive formula

In classical logic, disjunction is given a truth functional semantics according to which a formula

Disjunction has also been given numerous non-classical treatments, motivated by problems including Aristotle's sea battle argument, Heisenberg's uncertainty principle, as well as the numerous mismatches between classical disjunction and its nearest equivalents in natural languages.

When it is necessary to clarify whether inclusive or exclusive or is intended, English speakers sometimes uses the phrase and/or.

In terms of logic, this phrase is identical to or, but makes the inclusion of both being true explicit.

In logic and related fields, disjunction is customarily notated with an infix operator

In Jan Łukasiewicz's prefix notation for logic, the operator is

can be denoted as an iterated binary operation using a larger ⋁ (Unicode U+22C1 ⋁ N-ARY LOGICAL OR):[5]

Its semantic entry is standardly given as follows:[a] This semantics corresponds to the following truth table:[1] In classical logic systems where logical disjunction is not a primitive, it can be defined in terms of the primitive and (

) and not as:[6] The latter can be checked by the following truth table: It may also be defined solely in terms of

[citation needed] Many languages distinguish between bitwise and logical disjunction by providing two distinct operators; in languages following C, bitwise disjunction is performed with the single pipe operator (|), and logical disjunction with the double pipe (||) operator.

The logical disjunction operator thus usually constitutes a sequence point.

Although the type of a logical disjunction expression is Boolean in most languages (and thus can only have the value true or false), in some languages (such as Python and JavaScript), the logical disjunction operator returns one of its operands: the first operand if it evaluates to a true value, and the second operand otherwise.

The Curry–Howard correspondence relates a constructivist form of disjunction to tagged union types.

[11] Disjunction in natural languages does not precisely match the interpretation of

[1] This inference has sometimes been understood as an entailment, for instance by Alfred Tarski, who suggested that natural language disjunction is ambiguous between a classical and a nonclassical interpretation.

More recent work in pragmatics has shown that this inference can be derived as a conversational implicature on the basis of a semantic denotation which behaves classically.

However, disjunctive constructions including Hungarian vagy... vagy and French soit... soit have been argued to be inherently exclusive, rendering ungrammaticality in contexts where an inclusive reading would otherwise be forced.

As with exclusivity, these inferences have been analyzed both as implicatures and as entailments arising from a nonclassical interpretation of disjunction.

[1] In many languages, disjunctive expressions play a role in question formation.

For instance, while the above English example can be interpreted as a polar question asking whether it's true that Mary is either a philosopher or a linguist, it can also be interpreted as an alternative question asking which of the two professions is hers.

The role of disjunction in these cases has been analyzed using nonclassical logics such as alternative semantics and inquisitive semantics, which have also been adopted to explain the free choice and simplification inferences.

[1] In English, as in many other languages, disjunction is expressed by a coordinating conjunction.

In many languages such as Dyirbal and Maricopa, disjunction is marked using a verb suffix.

For instance, in the Maricopa example below, disjunction is marked by the suffix šaa.