Logical disjunction

OR
Definition	${\displaystyle x+y}$
Truth table	${\displaystyle (1110)}$
Logic gate
Normal forms
Disjunctive	${\displaystyle x+y}$
Conjunctive	${\displaystyle x+y}$
Zhegalkin polynomial	${\displaystyle x\oplus y\oplus xy}$
Post's lattices
0-preserving	yes
1-preserving	yes
Monotone	yes
Affine	no

Venn diagram of ${\displaystyle \scriptstyle A\lor B\lor C}$

In logic, disjunction, also known as logical disjunction or logical or or logical addition or inclusive disjunction , is a logical connective typically notated as ${\displaystyle \lor }$ 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 ${\displaystyle S\lor W}$, assuming that ${\displaystyle S}$ abbreviates "it is sunny" and ${\displaystyle W}$ abbreviates "it is warm".

In classical logic, disjunction is given a truth functional semantics according to which a formula ${\displaystyle \phi \lor \psi }$ is true unless both ${\displaystyle \phi }$ and ${\displaystyle \psi }$ 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. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. Disjunction has also been given numerous non-classical treatments, motivated by problems including Aristotle's sea battle argument, Heisenberg's uncertainty principle, as well as the numerous mismatches between classical disjunction and its nearest equivalents in natural languages.^[1][2]

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 ${\displaystyle \lor }$ (Unicode U+2228 ∨ LOGICAL OR).^[1] Alternative notations include ${\displaystyle +}$, used mainly in electronics, as well as ${\displaystyle \vert }$ and ${\displaystyle \vert \!\vert }$ 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 ${\displaystyle A}$, short for Polish alternatywa (English: alternative).^[3]

Classical disjunction

Semantics

In the semantics of logic, classical disjunction is a truth functional operation which returns the truth value "true" unless both of its arguments are "false". Its semantic entry is standardly given as follows:^[4]

${\displaystyle \models \phi \lor \psi }$     if     ${\displaystyle \models \phi }$     or     ${\displaystyle \models \psi }$     or     both

This semantics corresponds to the following truth table:^[1]

${\displaystyle A}$	${\displaystyle B}$	${\displaystyle A\lor B}$
True	True	True
True	False	True
False	True	True
False	False	False

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" (${\displaystyle \land }$) and "not" (${\displaystyle \lnot }$) as:

${\displaystyle A\lor B=\neg ((\neg A)\land (\neg B))}$.

Alternatively, it may be defined in terms of "implies" (${\displaystyle \to }$) and "not" as:^[5]

${\displaystyle A\lor B=(\lnot A)\to B}$.

The latter can be checked by the following truth table:

${\displaystyle A}$	${\displaystyle B}$	${\displaystyle \neg A}$	${\displaystyle \neg A\to B}$	${\displaystyle A\lor B}$
True	True	False	True	True
True	False	False	True	True
False	True	True	True	True
False	False	True	False	False

Applications in computer science

OR logic gate

Operators corresponding to logical disjunction exist in most programming languages.

Set theory

The membership of an element of a union set in set theory is defined in terms of a logical disjunction: ${\displaystyle x\in A\cup B\Leftrightarrow (x\in A)\vee (x\in B)}$. 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 ${\displaystyle \lor }$ in classical logic. Notably, classical disjunction is inclusive while natural language disjunction is often understood exclusively, as the following English 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 on the basis of a semantic denotation which behaves classically. However, disjunctive constructions including Hungarian vagy... vagy and French soit... 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 as free choice disjunction and simplification of disjunctive antecedents, 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.^[1]

• You can have an apple or a pear.

${\displaystyle \rightsquigarrow }$ 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 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.^[1]

• Is Mary a philosopher or a linguist?

In English, as in many other languages, disjunction is expressed by a coordinating conjunction. 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. For instance, in the Maricopa example below, disjunction is marked by the suffix šaa.^[1]

See also

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

Wikimedia Commons has media related to Logical disjunction.

• "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