Logical disjunction

In logic, disjunction is a logical connective typically notated formula_1 whose meaning either refines or corresponds to that of natural language expressions such as "or". In classical logic, it is given a truth functional semantics on which formula_2 is true unless both formula_3 and formula_4 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 the numerous mismatches between classical disjunction and its nearest equivalents in natural language.
Notation.
In logic and related fields, disjunction is customarily notated with an infix operator formula_5. Alternative notations include formula_6, used mainly in electronics, as well as formula_7 and formula_8 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 A, short for Polish "alternatywa" (English: alternative).
Classical disjunction.
Semantics.
Classical disjunction is a truth functional operation which returns the truth value "true" unless both of its arguments are "false". Its semantic entry is standardly given as follows:
This semantics is corresponds to the following truth table:
Defined by other operators.
In systems where logical disjunction is not a primitive, it may be defined as
This can be checked by the following truth table:
Properties.
The following properties apply to disjunction:
Applications in computer science.
Operators corresponding to logical disjunction exist in most programming languages.
Bitwise operation.
Disjunction is often used for bitwise operations. Examples:
The codice_1 operator can be used to set bits in a bit field to 1, by codice_1-ing the field with a constant field with the relevant bits set to 1. For example, codice_3 will force the final bit to 1, while leaving other bits unchanged.
Logical operation.
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 (codice_4), and logical disjunction with the double pipe (codice_5) operator.
Logical disjunction is usually short-circuited; that is, if the first (left) operand evaluates to codice_6, then the second (right) operand is not evaluated. The logical disjunction operator thus usually constitutes a sequence point.
In a parallel (concurrent) language, it is possible to short-circuit both sides: they are evaluated in parallel, and if one terminates with value true, the other is interrupted. This operator is thus called the parallel or.
Although the type of a logical disjunction expression is boolean in most languages (and thus can only have the value codice_6 or codice_8), 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.
Constructive disjunction.
The Curry–Howard correspondence relates a constructivist form of disjunction to tagged union types.
Set theory.
The membership of an element of a union set in set theory is defined in terms of a logical disjunction: formula_21 if and only if formula_22. Because of this, logical disjunction satisfies many of the same identities as set-theoretic union, such as associativity, commutativity, distributivity, and de Morgan's laws, identifying logical conjunction with set intersection, logical negation with set complement.
Natural language.
The classical denotation for formula_5 does not precisely match the denotation of disjunctive statements in natural languages such as English. Notably, classical disjunction is inclusive while natural language disjunction is often understood exclusively.
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.
Similar deviations from classical logic have been noted in cases such as free choice disjunction and simplification of disjunctive antecedents, where certain modal operators trigger a conjunction-like interpretation of disjunction. As with exclusivity, these inferences have been analyzed both as implicatures and as entailments arising from a nonclassical interpretation of disjunction.
In many languages, disjunctive expressions play a role in question formation. For instance, while the following 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 be analyzed using nonclassical logics such as alternative semantics and inquisitive semantics, which have also been adopted to explain the free choice and simplification inferences.
In English, as in many other languages, disjunction is expressed by a coordinating conjunction. Other languages express disjunctive meanings in a variety of ways, though it is unknown whether disjunction itself is a linguistic universal. 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".