"Disjunction" redirects here. For the logic gate, see OR gate. For separation of chromosomes, see Meiosis. For disjunctions in distribution, see Disjunct distribution.
"Logical OR" redirects here. For the operator ‖ as used in engineering and network theory, see Parallel addition (operator).
In logic, disjunction, also known as logical disjunction or logical or or logical addition or inclusive disjunction, is a logical connective typically notated as 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 , assuming that abbreviates "it is sunny" and abbreviates "it is warm".
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 are 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 uses the phrase "and/or". In terms of logic, this phrase is identical to "or", but makes the inclusion of both being true explicit.
Notation
In logic and related fields, disjunction is customarily notated with an infix operator (Unicode U+2228∨LOGICAL OR).[1] Alternative notations include , used mainly in electronics, as well as and 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 , short for Polish alternatywa (English: alternative).[4]
Truth-preserving: The interpretation under which all variables are assigned a truth value of 'true', produces a truth value of 'true' as a result of disjunction.
Falsehood-preserving: The interpretation under which all variables are assigned a truth value of 'false', produces a truth value of 'false' as a result of disjunction.
Applications in computer science
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The or operator can be used to set bits in a bit field to 1, by or-ing the field with a constant field with the relevant bits set to 1. For example, x = x | 0b00000001 will force the final bit to 1, while leaving other bits unchanged.[citation needed]
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 (|), and logical disjunction with the double pipe (||) operator.
Logical disjunction is usually short-circuited; that is, if the first (left) operand evaluates to true, 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 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.[8][9] This allows it to fulfill the role of the Elvis operator.
Disjunction in natural languages does not precisely match the interpretation of in classical logic. Notably, classical disjunction is inclusive while natural language disjunction is often understood exclusively, as the following English example typically would be.[1]
Mary is eating an apple or a pear.
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 Hungarianvagy... vagy and Frenchsoit... soit have been argued to be inherently exclusive, rendering ungrammaticality in contexts where an inclusive reading would otherwise be forced.[1]
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.[1]
You can have an apple or a pear.
You can have an apple and you can have a pear (but you can't have both)
In many languages, disjunctive expressions play a role in question formation.
Is Mary a philosopher or a linguist?
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. 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.[1]
George Boole, closely following analogy with ordinary mathematics, premised, as a necessary condition to the definition of "x + y", that x and y were mutually exclusive. Jevons, and practically all mathematical logicians after him, advocated, on various grounds, the definition of "logical addition" in a form which does not necessitate mutual exclusiveness.
References
^ abcdefghAloni, Maria (2016), "Disjunction", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2016 ed.), Metaphysics Research Lab, Stanford University, retrieved 2020-09-03
^Beall, Jeffrey C. (2010). Logic: the basics. The basics (1. publ ed.). London: Routledge. p. 57. ISBN978-0-203-85155-5.
^Józef Maria Bocheński (1959), A Précis of Mathematical Logic, translated by Otto Bird from the French and German editions, Dordrecht, North Holland: D. Reidel, passim.
^For the sake of generality across classical systems, this entry suppresses the parameters of evaluation. The "double turnstile" symbol here is intended to mean "semantically entails".
^Marcus Vinícius Midena Ramos; de Queiroz, Ruy J. G. B. (2015). "Context-Free Language Theory Formalization". Universidade Federal de Pernambuco: 6. arXiv:1505.00061.
^Ebbinghaus, Heinz-Dieter (2021). Einführung in die Mengenlehre (in German) (5 ed.). Springer. p. 32. ISBN978-3-662-63865-1.