In
logic, disjunction is a
logical connective 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, 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
exclusive disjunction. 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, as well the numerous mismatches between classical disjunction and its nearest equivalents in
natural languages.
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, 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, as well as
and
in many
programming languages. 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 functional
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 "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 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:
*
Commutativity:
*
Distributivity:
:::
:::
:::
*
Idempotency:
*
Monotonicity:
:::
*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
Operators corresponding to logical disjunction exist in most
programming languages.
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.
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 relates a
constructivist form of disjunction to
tagged union types.
Set theory
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 set-theoretic union, such as
associativity,
commutativity,
distributivity, and
de Morgan's laws, identifying
logical conjunction with
set intersection,
logical negation with
set complement.
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
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 between a classical and a nonclassical interpretation. More recent work in
pragmatics 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 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 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
*
Boolean algebra (logic)
*
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
*
Propositional calculus
*
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, 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
Semantics
Formal semantics (natural language)