Logical disjunction

In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as ${\displaystyle \lor }$ and read aloud as "or". For instance, the English language sentence "it is sunny or it is warm" can be represented in logic using the disjunctive formula ${\displaystyle S\lor W}$, assuming that ${\displaystyle S}$ abbreviates "it is sunny" and ${\displaystyle W}$ abbreviates "it is warm".

In classical logic, disjunction is given a truth functional semantics according to which a formula ${\displaystyle \phi \lor \psi }$ is true unless both ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an inclusive interpretation of disjunction, in contrast with exclusive disjunction. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. 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.

An operand of a disjunction is a disjunct.

Because the logical or means a disjunction formula is true when either one or both of its parts are true, it is referred to as an inclusive disjunction. This is in contrast with an exclusive disjunction, which is true when one or the other of the arguments is true, but not both (referred to as exclusive or, or XOR).

When it is necessary to clarify whether inclusive or exclusive or is intended, English speakers sometimes use 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 ${\displaystyle \lor }$ (Unicode U+2228 ∨ LOGICAL OR). Alternative notations include ${\displaystyle +}$, used mainly in electronics, as well as ${\displaystyle \vert }$ and ${\displaystyle \vert \!\vert }$ in many programming languages. The English word or is sometimes used as well, often in capital letters. In Jan Łukasiewicz's prefix notation for logic, the operator is ${\displaystyle A}$, short for Polish alternatywa (English: alternative).

In mathematics, the disjunction of an arbitrary number of elements ${\displaystyle a_{1},\ldots ,a_{n}}$ can be denoted as an iterated binary operation using a larger ⋁ (Unicode U+22C1 ⋁ N-ARY LOGICAL OR):

${\displaystyle \bigvee _{i=1}^{n}a_{i}=a_{1}\lor a_{2}\lor \ldots a_{n-1}\lor a_{n}}$