In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, disjunction is a
logical connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
typically notated as
and read aloud as "or". For instance, the
English
English usually refers to:
* English language
* English people
English may also refer to:
Peoples, culture, and language
* ''English'', an adjective for something of, from, or related to England
** English national ...
language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula
, assuming that
abbreviates "it is raining" and
abbreviates "it is snowing".
In
classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...
, disjunction is given a
truth function
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly o ...
al 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
exclusive disjunction
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...
. 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 of propositional logic and almost every other deduction system. The rule makes it possible to introduce disjunctions to logical proofs. It is the infer ...
and
disjunction elimination. 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 ''contingent:'' neither necessarily true nor necessarily false.
The problem of future contingents seems to have been firs ...
,
Heisenberg's
uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
, as well the numerous mismatches between classical disjunction and its nearest equivalents in
natural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...
s.
Inclusive and exclusive disjunction
Because the logical "or" means a formula is when either or both are true, it is referred to as an ''inclusive'' disjunction. This is in contrast with an
exclusive disjunction
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...
, 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
.
Alternative notations include
, used mainly in
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
, as well as
and
in many
programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...
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 His work centred on philosophical logic, mathematical logic and history of logic. ...
's
prefix notation for logic, the operator is A, short for Polish ''alternatywa'' (English: alternative).
Classical disjunction
Semantics
In the
semantics of logic, classical disjunction is a
truth function
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly o ...
al
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
which returns the
truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
Computing
In some pro ...
"true" unless both of its arguments are "false". Its semantic entry is standardly given as follows:
::
if
or
or both
This semantics corresponds to the following
truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...
:
Defined by other operators
In
classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...
systems where logical disjunction is not a primitive, it can be defined in terms of the primitive "
and
or AND may refer to:
Logic, grammar, and computing
* Conjunction (grammar), connecting two words, phrases, or clauses
* Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition
* Bitwise AND, a boolea ...
" (
) and "
not" (
) as:
:
.
Alternatively, it may be defined in terms of "
implies" (
) and "not" as:
:
.
The latter 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 valid rule of replacement ...
:
*
Commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
:
*
Distributivity:
:::
:::
:::
*
Idempotency:
*
Monotonicity:
:::
*Truth-preserving: The interpretation under which all variables are assigned a
truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
Computing
In some pro ...
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 and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
Computing
In some pro ...
of 'false', produces a truth value of 'false' as a result of disjunction.
Applications in computer science
Operators corresponding to logical disjunction exist in most
programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...
s.
Bitwise operation
Disjunction is often used for
bitwise operation
In computer programming, a bitwise operation operates on a bit string, a bit array or a 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-level arithmetic oper ...
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 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 and
JavaScript
JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web, alongside HTML and CSS. As of 2022, 98% of websites use JavaScript on the client side for webpage behavior, of ...
), 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
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct rela ...
relates a
constructivist form of disjunction to
tagged union types.
Set theory
The
membership of an element of a
union set in
set theory
Set theory is the branch of mathematical logic that studies 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, is mostly concern ...
is defined in terms of a logical disjunction:
. 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 valid rule of replacement ...
,
commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
,
distributivity, and
de Morgan's laws, identifying
logical conjunction
In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents thi ...
with
set intersection,
logical negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
with
set complement.
Natural language
Disjunction in
natural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...
s 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
English usually refers to:
* English language
* English people
English may also refer to:
Peoples, culture, and language
* ''English'', an adjective for something of, from, or related to England
** English national ...
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
Ambiguity is the 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. It is thus an attribute of any idea or statement ...
between a classical and a nonclassical interpretation. More recent work in
pragmatics
In linguistics and related fields, pragmatics is the study of how context contributes to meaning. The field of study evaluates how human language is utilized in social interactions, as well as the relationship between the interpreter and the int ...
has shown that this inference can be derived as a
conversational implicature
In pragmatics, a subdiscipline of linguistics, 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 sayi ...
on the basis of a
semantic
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
denotation which behaves classically. However, disjunctive constructions including
Hungarian ''vagy... vagy'' and
French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
** French language, which originated in France, and its various dialects and accents
** French people, a nation and ethnic group identified with Franc ...
''soit... soit'' have been argued to be inherently exclusive, rendering un
grammaticality
In linguistics, grammaticality is determined by the conformity to language usage as derived by the grammar of a particular speech variety. The notion of grammaticality rose alongside the theory of generative grammar, the goal of which is to form ...
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 In formal semantics and philosophical logic, simplification of disjunctive antecedents (SDA) is the phenomenon whereby a disjunction in the antecedent of a conditional appears to distribute over the conditional as a whole. This inference is shown ...
, 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.
:2. 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. 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 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.
: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 (abbreviated or ) is a part of speech that connects words, phrases, or clauses that are called the conjuncts of the conjunctions. That definition may overlap with that of other parts of speech and so what constitutes ...
. 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
In linguistics, a suffix is an affix which is placed after the stem of a word. Common examples are case endings, which indicate the grammatical case of nouns, adjectives, and verb endings, which form the conjugation of verbs. Suffixes can carr ...
. For instance, in the Maricopa example below, disjunction is marked by the suffix ''šaa''.
See also
*
Affirming a disjunct
*
Bitwise OR
In computer programming, a bitwise operation operates on a bit string, a bit array or a 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-level arithmetic oper ...
*
Boolean algebra (logic)
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
*
Boolean algebra topics
This is a list of topics around Boolean algebra and propositional logic.
Articles with a wide scope and introductions
* Algebra of sets
* Boolean algebra (structure)
* Boolean algebra
* Field of sets
* Logical connective
* Propo ...
*
Boolean domain
*
Boolean function
*
Boolean-valued function
*
Disjunctive syllogism
In classical logic, disjunctive syllogism (historically known as ''modus tollendo ponens'' (MTP), Latin for "mode that affirms by denying") is a valid argument form which is a syllogism having a disjunctive statement for one of its premises ...
*
Disjunction elimination
*
Disjunction introduction
Disjunction introduction or addition (also called or introduction) is a rule of inference of propositional logic and almost every other deduction system. The rule makes it possible to introduce disjunctions to logical proofs. It is the infer ...
*
First-order logic
*
Fréchet inequalities In probabilistic logic, the Fréchet inequalities, also known as the Boole–Fréchet inequalities, are rules implicit in the work of George BooleBoole, G. (1854). ''An Investigation of the Laws of Thought, On Which Are Founded the Mathematical Theo ...
*
Free choice inference
*
Hurford disjunction In formal semantics, a Hurford disjunction is a disjunction in which one of the disjuncts entails the other. The concept was first identified by British linguist James Hurford. The sentence "Mary is in the Netherlands or she is in Amsterdam" is a ...
*
Logical graph A logical graph is a special type of diagrammatic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic.
In his papers on ''qualitative logic'', ''entitative graphs'', and '' existential grap ...
*
Logical value
*
Operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
*
Operator (programming)
*
OR gate
The OR gate is a digital logic gate that implements logical disjunction. The OR gate returns true if either or both of its inputs are true; otherwise it returns false. The input and output states are normally represented by different voltage lev ...
*
Propositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
*
Simplification of disjunctive antecedents In formal semantics and philosophical logic, simplification of disjunctive antecedents (SDA) is the phenomenon whereby a disjunction in the antecedent of a conditional appears to distribute over the conditional as a whole. This inference is shown ...
Notes
*
George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in ...
, 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
External links
*
*
*Eric W. Weisstein
"Disjunction."From MathWorld—A Wolfram Web Resource
{{Authority control
Disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor ...
Semantics
Formal semantics (natural language)