In

"Disjunction."

From MathWorld—A Wolfram Web Resource {{Authority control

logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

, disjunction is a logical connective
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

typically notated $\backslash lor$ whose meaning either refines or corresponds to that of natural language expressions such as "or". In classical logic Classical logic (or standard logic) is the intensively studied and most widely used class of deductive logic
Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusio ...

, it is given a truth function
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

al semantics
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference
Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another o ...

on which $\backslash phi\; \backslash lor\; \backslash psi$ is true unless both $\backslash phi$ and $\backslash psi$ 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
Exclusive or or exclusive disjunction is a logical operation
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic ...

. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction
Disjunction introduction or addition (also called or introduction) is a rule of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, ana ...

and disjunction elimination
In propositional logic
Propositional calculus is a branch of logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Al ...

. Disjunction has also been given numerous non-classical treatments, motivated by problems including Aristotle's sea battle argument
Future contingent propositions (or simply, future contingents) are statements about states of affairs in the future that are ''contingency (philosophy), contingent:'' neither necessarily true nor necessarily false.
The problem of future contingents ...

, Heisenberg
Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the key pioneers of quantum mechanics
Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a de ...

's uncertainty principle
In quantum mechanics
Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quant ...

, 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 $\backslash lor$. Alternative notations include $+$, used mainly inelectronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons
The electron is a subatomic particle
In physical sciences, subatomic particles are smaller than ...

, as well as $\backslash vert$ and $\backslash vert\backslash !\backslash vert$ in many programming language
A programming language is a formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

s. The English word "or" is sometimes used as well, often in capital letters. In Jan Łukasiewicz
Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. He was born in Lemberg, a city in the Austrian Galicia, Galician Kingdom of Austria-Hungar ...

's prefix notation for logic, the operator is A, short for Polish ''alternatywa'' (English: alternative).
Classical disjunction

Semantics

Classical disjunction is atruth function
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

al operation which returns the truth value
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

"true" unless both of its arguments are "false". Its semantic entry is standardly given as follows:
:: $\backslash models\; \backslash phi\; \backslash lor\; \backslash psi$ if $\backslash models\; \backslash phi$ or $\backslash models\; \backslash psi$ or both
This semantics corresponds to the following truth table
A truth table is a mathematical table
Mathematical tables are lists of numbers showing the results of a calculation with varying arguments. Tables of trigonometric functions were used in ancient Greece and India for applications to astronomy
...

:
Defined by other operators

In systems where logical disjunction is not a primitive, it may be defined as :$A\; \backslash lor\; B\; =\; \backslash neg\; A\; \backslash to\; B$. This can be checked by the following truth table:Properties

The following properties apply to disjunction: *Associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...

: $a\; \backslash lor\; (b\; \backslash lor\; c)\; \backslash equiv\; (a\; \backslash lor\; b)\; \backslash lor\; c$
*Commutativity
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

: $a\; \backslash lor\; b\; \backslash equiv\; b\; \backslash lor\; a$
*Distributivity
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

: $(a\; \backslash land\; (b\; \backslash lor\; c))\; \backslash equiv\; ((a\; \backslash land\; b)\; \backslash lor\; (a\; \backslash land\; c))$
:::$(a\; \backslash lor\; (b\; \backslash land\; c))\; \backslash equiv\; ((a\; \backslash lor\; b)\; \backslash land\; (a\; \backslash lor\; c))$
:::$(a\; \backslash lor\; (b\; \backslash lor\; c))\; \backslash equiv\; ((a\; \backslash lor\; b)\; \backslash lor\; (a\; \backslash lor\; c))$
:::$(a\; \backslash lor\; (b\; \backslash equiv\; c))\; \backslash equiv\; ((a\; \backslash lor\; b)\; \backslash equiv\; (a\; \backslash lor\; c))$
*Idempotency
Idempotence (, ) is the property of certain operations in mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...

: $a\; \backslash lor\; a\; \backslash equiv\; a$
*Monotonicity
Figure 3. A function that is not monotonic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...

: $(a\; \backslash rightarrow\; b)\; \backslash rightarrow\; ((c\; \backslash lor\; a)\; \backslash rightarrow\; (c\; \backslash lor\; b))$
:::$(a\; \backslash rightarrow\; b)\; \backslash rightarrow\; ((a\; \backslash lor\; c)\; \backslash rightarrow\; (b\; \backslash lor\; c))$
*Truth-preserving: The interpretation under which all variables are assigned a truth value
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

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
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

of 'false', produces a truth value of 'false' as a result of disjunction.
Applications in computer science

Operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...

corresponding to logical disjunction exist in most programming language
A programming language is a formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

s.
Bitwise operation

Disjunction is often used forbitwise operation
In computer programming, a bitwise operation operates on a bit string, a bit array or a Binary numeral system, binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher le ...

s. 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
A bit field is a data structure
Image:Hash table 3 1 1 0 1 0 0 SP.svg, 315px, A data structure known as a hash table.
In computer science, a data structure is a data organization, management, and storage format that enables efficient access and ...

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.
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
PYTHON was a Cold War contingency plan of the Government of the United Kingdom, British Government for the continuity of government in the event of Nuclear warfare, nuclear war.
Background
Following the report of the Strath Committee in 1955, the ...

and JavaScript
JavaScript (), often abbreviated JS, is a programming language
A programming language is a formal language
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), ma ...

), 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

TheCurry–Howard correspondence
In programming language theory and proof theory
Proof may refer to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Formal sciences
* Formal proof, a c ...

relates a constructivist form of disjunction to tagged unionIn computer science, a tagged union, also called a variant type, variant, variant record, choice type, discriminated union, disjoint union, sum type or coproduct, is a data structure used to hold a value that could take on several different, but fixe ...

types.
Set theory

Themembership
Member may refer to:
* Military juryA United States military "jury" (or "Members", in military parlance) serves a function similar to an American civilian jury, but with several notable differences. Only a Courts-martial in the United States, Gene ...

of an element of a union set in set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...

is defined in terms of a logical disjunction: $x\backslash in\; A\backslash cup\; B$ if and only if $(x\backslash in\; A)\backslash vee(x\backslash in\; B)$. Because of this, logical disjunction satisfies many of the same identities as set-theoretic union, such as associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...

, commutativity
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, distributivity
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, and de Morgan's laws
In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are named after Augustus De Morgan, a 19th ...

, identifying logical conjunction
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...

with set intersection
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, logical negation
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argume ...

with set complement
In , the complement of a , often denoted by (or ), are the not in .
When all sets under consideration are considered to be s of a given set , the absolute complement of is the set of elements in that are not in .
The relative complement of ...

.
Natural language

The classical denotation for $\backslash lor$ does not precisely match thedenotation
The denotation of a word is its central sense
A sense is a biological system used by an organism for sensation, the process of gathering information about the world and responding to Stimulus (physiology), stimuli. (For example, in the human bod ...

of disjunctive statements in natural language
In neuropsychology
Neuropsychology is a branch of psychology. It is concerned with how a person's cognition and behavior are related to the brain and the rest of the nervous system. Professionals in this branch of psychology often focus on ...

s such as English
English usually refers to:
* English language
English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become the World language, leading lan ...

. Notably, classical disjunction is inclusive while natural language disjunction is often understood exclusively.
:1. Mary is patriotic or quixotic.
::$\backslash rightsquigarrow$ Mary is not both patriotic and quixotic.
This inference has sometimes been understood as an entailment
Logical consequence (also entailment) is a fundamental concept
Concepts are defined as abstract ideas
A mental representation (or cognitive representation), in philosophy of mind
Philosophy of mind is a branch of philosophy that studies th ...

, for instance by Alfred Tarski
Alfred Tarski (; January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. was a Polish-American logician ...

, who suggested that natural language disjunction is ambiguous
Ambiguity is a type of meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations plausible. A common aspect of ambiguity is uncertainty
Uncertainty refers to Epistemology, epistemic sit ...

between a classical and a nonclassical interpretation. More recent work in pragmatics
In linguistics
Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the m ...

has shown that this inference can be derived as a conversational implicature
An implicature is something the speaker suggests or implies with an utterance, even though it is not literally expressed. Implicatures can aid in communicating more efficiently than by explicitly saying everything we want to communicate. This phenom ...

on the basis of a semantic
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference
Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another ...

denotation which behaves classically. However, disjunctive constructions including HungarianHungarian may refer to:
* Hungary, a country in Central Europe
* Kingdom of Hungary, state of Hungary, existing between 1000 and 1946
* Hungarians, ethnic groups in Hungary
* Hungarian algorithm, a polynomial time algorithm for solving the assignmen ...

''vagy... vagy'' and ''soit... soit'' have been argued to be inherently exclusive, rendering ungrammaticality
In linguistics
Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them.
The traditional areas of linguistic analysis include ...

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
Conjunction may refer to:
* Conjunction (astronomy), in which two astronomical bodies appear close together in the sky
* Conjunction (astrology), astrological aspect in horoscopic astrology
* Conjunction (grammar), a part of speech
* Logical conjun ...

-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.
:2. You can have an apple or a pear.
::$\backslash rightsquigarrow$ 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. For instance, while the following English example can be interpreted as a polar question
Polar may refer to:
Geography
Polar may refer to:
* Geographical pole
A geographical pole or geographic pole is either of the two points on Earth where its axis of rotation intersects its surface. The North Pole lies in the Arctic Ocean
...

asking whether it's true that Mary is either a philosopher or a linguist, it can also be interpreted as an alternative question
''AlterNative: An International Journal of Indigenous Peoples'' (formerly ''AlterNative: An International Journal of Indigenous Scholarship'') is a quarterly peer-reviewed academic journal
An academic or scholarly journal is a periodical publica ...

asking which of the two professions is hers. The role of disjunction in these cases has be analyzed using nonclassical logics such as alternative semanticsAlternative semantics (or Hamblin semantics) is a framework in formal semantics and logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, t ...

and inquisitive semantics
Inquisitive semantics is a framework in logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos' ...

, which have also been adopted to explain the free choice and simplification inferences.
:3. Is Mary a philosopher or a linguist?
In English, as in many other languages, disjunction is expressed by a coordinating conjunction
In grammar, a conjunction (list of glossing abbreviations, abbreviated or ) is a part of speech that connects words, phrases, or clauses that are called the conjuncts of the conjunctions. This definition may overlap with that of other parts of s ...

. Other languages express disjunctive meanings in a variety of ways, though it is unknown whether disjunction itself is a linguistic universal
A linguistic universal is a pattern that occurs systematically across natural languages, potentially true for all of them. For example, ''All languages have noun
A noun (from Latin ''nōmen'', literally ''name'') is a word that functions as the ...

. In many languages such as Dyirbal and Maricopa
Maricopa can refer to:
Places
* Maricopa, Arizona, United States, a city
** Maricopa Freeway, a piece of I-10 in Metropolitan Phoenix
** Maricopa station, an Amtrak station in Maricopa, Arizona
* Maricopa County, Arizona, United States
* Maricopa ...

, disjunction is marked using a verb suffix
In linguistics
Linguistics is the scientific study of language
A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most languag ...

. For instance, in the Maricopa example below, disjunction is marked by the suffix ''šaa''.
See also

*Affirming a disjunct
The formal fallacy of affirming a disjunct also known as the fallacy of the alternative disjunct or a false exclusionary disjunct occurs when a deductive logic, deductive argument takes the following logical form:
:A or B
:A
:Therefore, not B
...

* Bitwise OR
In computer programming
Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a specific task. Programming involves tasks such as: analysis, gener ...

* Boolean algebra (logic)
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, respectively. Instead of elementary a ...

* Boolean algebra topics
* Boolean domain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

* Boolean function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

* Boolean-valued function
A Boolean-valued function (sometimes called a predicate or a proposition
In linguistics and logic, a proposition is the meaning of a declarative sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic enti ...

* Disjunctive syllogism
In classical logicClassical logic (or standard logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-s ...

* Disjunction elimination
In propositional logic
Propositional calculus is a branch of logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Al ...

* Disjunction introduction
Disjunction introduction or addition (also called or introduction) is a rule of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, ana ...

* First-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system
A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...

* Fréchet inequalitiesIn probabilistic logicThe aim of a probabilistic logic (also probability logic and probabilistic reasoning) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure of formal pr ...

* Free choice inferenceFree choice is a phenomenon in natural language where a disjunction
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=lo ...

* Hurford disjunction
* Logical graph
* Logical value
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

* Operation
* Operator (programming)
In computer programming
Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, ge ...

* OR gate
The OR gate is a digital logic gate that implements logical disjunctionit behaves according to the truth table to the right. A HIGH output (1) results if one or both the inputs to the gate are HIGH (1). If neither input is high, a LOW output ( ...

* Propositional calculus
Propositional calculus is a branch of logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...

* Simplification of disjunctive antecedents
Notes

*George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught
Autodidacticism (also autodidactism) or self-education (also self-learning and self-teaching) is education
Education is the process of facil ...

, closely following analogy with ordinary mathematics, premised, as a necessary condition to the definition of "x + y", that x and y were mutually exclusive. , 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

External links

* * *Eric W. Weisstein"Disjunction."

From MathWorld—A Wolfram Web Resource {{Authority control

Disjunction
In logic, disjunction is a logical connective typically notated \lor 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 of logic, sema ...

Semantics
Formal semantics (natural language)