Logical disjunction
OR  

Definition  
Truth table  
Logic gate  
Normal forms  
Disjunctive  
Conjunctive  
Zhegalkin polynomial  
Post's lattices  
0preserving  yes 
1preserving  yes 
Monotone  yes 
Affine  no 
Selfdual  no 
Logical connectives  



Related concepts  
Applications  
Category  
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".
In classical logic, disjunction is given a truth 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 an inclusive interpretation of disjunction, in contrast with
An operand of a disjunction is a disjunct.^{[3]}
Inclusive and exclusive disjunction
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]}
In mathematics, the disjunction of an arbitrary number of elements can be denoted as an iterated binary operation using a larger ⋁ (Unicode U+22C1 ⋁ NARY LOGICAL OR):^{[5]}
Classical disjunction
Semantics
In the
 if or or both
This semantics corresponds to the following truth table:^{[1]}
F  F  F 
F  T  T 
T  F  T 
T  T  T 
Defined by other operators
In classical logic systems where logical disjunction is not a primitive, it can be defined in terms of the primitive "and" () and "
 .
Alternatively, it may be defined in terms of "implies" () and "not" as:^{[7]}
 .
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 
Properties
The following properties apply to disjunction:
 Associativity: ^{[8]}
 Commutativity:
 Distributivity:
 Idempotency:
 Monotonicity:
 Truthpreserving: The interpretation under which all variables are assigned a truth value of 'true', produces a truth value of 'true' as a result of disjunction.
 Falsehoodpreserving: 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
This section needs expansion. You can help by adding to it. (February 2021) 
Bitwise operation
Disjunction is often used for bitwise operations. Examples:
 0 or 0 = 0
 0 or 1 = 1
 1 or 0 = 1
 1 or 1 = 1
 1010 or 1100 = 1110
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

), and logical disjunction with the double pipe (
) operator.
Logical disjunction is usually shortcircuited; 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 shortcircuit 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
Constructive disjunction
The
Set theory
This section needs expansion. You can help by adding to it. (February 2021) 
The membership of an element of a union set in set theory is defined in terms of a logical disjunction: . Because of this, logical disjunction satisfies many of the same identities as settheoretic union, such as
Natural language
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
Similar deviations from classical logic have been noted in cases such as
 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
In English, as in many other languages, disjunction is expressed by a
Johnš
JohnNOM
Billš
BillNOM
vʔaawuumšaa
3comePLFUTINFER
'John or Bill will come.'
See also
 Affirming a disjunct
 Boolean algebra (logic)
 Boolean algebra topics
 Boolean domain
 Boolean function
 Booleanvalued function
 Conjunction/disjunction duality
 Disjunctive syllogism
 Fréchet inequalities
 Free choice inference
 Hurford disjunction
 Logical graph
 Simplification of disjunctive antecedents
Notes
 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
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} Aloni, Maria (2016), "Disjunction", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2016 ed.), Metaphysics Research Lab, Stanford University, retrieved 20200903
 ^ "Disjunction  logic". Encyclopedia Britannica. Retrieved 20200903.
 ISBN 9780203851555.
 ^ 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.
 ^ Weisstein, Eric W. "OR". MathWorldA Wolfram Web Resource. Retrieved 24 September 2024.
 ^ 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".
 ISBN 9789814343879.
 .
 ^ "Python 3.12.1 Documentation  The Python Language Reference  6.11 Boolean operations". Retrieved 25 Dec 2023.
 ^ "JavaScript References  Expressions & Operators  Logical AND (&&)". 25 September 2023. Retrieved 25 Dec 2023.
 arXiv:1505.00061.
 .
External links
 "Disjunction", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 Aloni, Maria. "Disjunction". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
 Eric W. Weisstein. "Disjunction." From MathWorld—A Wolfram Web Resource