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| Disjunctive | |
| Conjunctive | |
| Zhegalkin polynomial | |
| Post's lattices | |
| 0-preserving | yes |
| 1-preserving | yes |
| Monotone | yes |
| Affine | no |
| Self-dual | no |
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Inlogic,disjunction, also known aslogical disjunction orlogical or orlogical addition orinclusive disjunction, is alogical connective typically notated as and read aloud as "or". For instance, theEnglish 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".
Inclassical logic, disjunction is given atruth functional semantics according to which a formula is true unless both and are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is aninclusive interpretation of disjunction, in contrast withexclusive disjunction. Classicalproof theoretical treatments are often given in terms of rules such asdisjunction introduction anddisjunction elimination. Disjunction has also been given numerousnon-classical treatments, motivated by problems includingAristotle's sea battle argument,Heisenberg'suncertainty principle, as well as the numerous mismatches between classical disjunction and its nearest equivalents innatural languages.[1][2]
Anoperand of a disjunction is adisjunct.[3]
Because the logicalor means a disjunction formula is true when either one or both of its parts are true, it is referred to as aninclusive disjunction. This is in contrast with anexclusive disjunction, which is true when one or the other of the arguments are true, but not both (referred to asexclusive or, orXOR).
When it is necessary to clarify whether inclusive or exclusiveor is intended, English speakers sometimes uses the phraseand/or. In terms of logic, this phrase is identical toor, but makes the inclusion of both being true explicit.
In logic and related fields, disjunction is customarily notated with an infix operator (UnicodeU+2228 ∨LOGICAL OR).[1] Alternative notations include, used mainly inelectronics, as well as and in manyprogramming languages. The English wordor is sometimes used as well, often in capital letters. InJan Łukasiewicz'sprefix notation for logic, the operator is, short for Polishalternatywa (English: alternative).[4]
In mathematics, the disjunction of an arbitrary number of elements can be denoted as aniterated binary operation using a larger ⋁ (UnicodeU+22C1 ⋁N-ARY LOGICAL OR):[5]
In thesemantics of logic, classical disjunction is atruth functionaloperation which returns thetruth valuetrue unless both of its arguments arefalse. Its semantic entry is standardly given as follows:[a]
This semantics corresponds to the followingtruth table:[1]
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
Inclassical logic systems where logical disjunction is not a primitive, it can be defined in terms of the primitiveand () andnot () as:
Alternatively, it may be defined in terms ofimplies () andnot as:[6]
The latter can be checked by the following truth table:
| F | F | T | F | F |
| F | T | T | T | T |
| T | F | F | T | T |
| T | T | F | T | T |
It may also be defined solely in terms of:
It can be checked by the following truth table:
| F | F | T | F | F |
| F | T | T | T | T |
| T | F | F | T | T |
| T | T | T | T | T |
The following properties apply to disjunction:
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Operators corresponding to logical disjunction exist in mostprogramming languages.
Disjunction is often used forbitwise operations. Examples:
Theor operator can be used to set bits in abit field to 1, byor-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]
Many languages distinguish between bitwise and logical disjunction by providing two distinct operators; in languages followingC,bitwise disjunction is performed with the single pipe operator (|), and logical disjunction with the double pipe (||) operator.
Logical disjunction is usuallyshort-circuited; that is, if the first (left) operand evaluates totrue, then the second (right) operand is not evaluated. The logical disjunction operator thus usually constitutes asequence 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 theparallel or.
Although the type of a logical disjunction expression is Boolean in most languages (and thus can only have the valuetrue orfalse), in some languages (such asPython andJavaScript), 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 theElvis operator.
TheCurry–Howard correspondence relates aconstructivist form of disjunction totagged union types.[citation needed][10]
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Themembership of an element of aunion set inset theory is defined in terms of a logical disjunction:. Because of this, logical disjunction satisfies many of the same identities as set-theoretic union, such asassociativity,commutativity,distributivity, andde Morgan's laws, identifyinglogical conjunction withset intersection,logical negation withset complement.[11]
Disjunction innatural 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]
This inference has sometimes been understood as anentailment, for instance byAlfred Tarski, who suggested that natural language disjunction isambiguous between a classical and a nonclassical interpretation. More recent work inpragmatics has shown that this inference can be derived as aconversational implicature on the basis of asemantic denotation which behaves classically. However, disjunctive constructions includingHungarianvagy... vagy andFrenchsoit... 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 asfree choice disjunction andsimplification of disjunctive antecedents, where certainmodal operators trigger aconjunction-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]
In many languages, disjunctive expressions play a role in question formation.
For instance, while the above English example can be interpreted as apolar question asking whether it's true that Mary is either a philosopher or a linguist, it can also be interpreted as analternative question asking which of the two professions is hers. The role of disjunction in these cases has been analyzed using nonclassical logics such asalternative semantics andinquisitive 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 acoordinating conjunction. Other languages express disjunctive meanings in a variety of ways, though it is unknown whether disjunction itself is alinguistic universal. In many languages such asDyirbal andMaricopa, disjunction is marked using a verbsuffix. For instance, in the Maricopa example below, disjunction is marked by the suffixšaa.[1]
Johnš
John-NOM
Billš
Bill-NOM
Johnš Billš vʔaawuumšaa
John-NOM Bill-NOM 3-come-PL-FUT-INFER
'John or Bill will come.'
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