Logical disjunction

 ( Disjunction ) 

In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as $\backslash lor$ and read aloud as "or". For instance, the English language language sentence "it is sunny or it is warm" can be represented in logic using the disjunctive formula $S\; \backslash lor\; W$, assuming that $S$ abbreviates "it is sunny" and $W$ abbreviates "it is warm".

In classical logic, disjunction is given a semantics according to which a formula $\backslash phi\; \backslash lor\; \backslash psi$ 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. Classical proof theory 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 .

An operand of a disjunction is a disjunct.

---

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 $\backslash lor$ (Unicode ). Alternative notations include $+$, used mainly in electronics, as well as $\backslash vert$ and $\backslash vert\backslash !\backslash vert$ 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 $A$, short for Polish alternatywa (English: alternative).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.

In mathematics, the disjunction of an arbitrary number of elements $a\_1,\; \backslash ldots,\; a\_n$ can be denoted as an iterated binary operation using a larger ⋁ (Unicode ):

$\backslash bigvee\_\{i=1\}^\{n\}\; a\_i\; =\; a\_1\; \backslash lor\; a\_2\; \backslash lor\; \backslash ldots\; a\_\{n-1\}\; \backslash lor\; a\_\{n\}$

---

Classical disjunction

---

Semantics

In the semantics of logic, classical disjunction is a operation which returns the truth value 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:

---

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 ($\backslash land$) and logical negation ($\backslash lnot$) as:

$A\; \backslash lor\; B\; =\; \backslash neg\; ((\backslash neg\; A)\; \backslash land\; (\backslash neg\; B))$.

Alternatively, it may be defined in terms of implies ($\backslash to$) and not as:

Michał Walicki (2025). 9789814343879, WORLD SCIENTIFIC. . ISBN 9789814343879

$A\; \backslash lor\; B\; =\; (\backslash lnot\; A)\; \backslash to\; B$.

The latter can be checked by the following truth table:

It may also be defined solely in terms of $\backslash to$:

$A\; \backslash lor\; B\; =\; (A\; \backslash to\; B)\; \backslash to\; B$.

It can be checked by the following truth table:

---

Properties The following properties apply to disjunction:

• Associativity: $a\; \backslash lor\; (b\; \backslash lor\; c)\; \backslash equiv\; (a\; \backslash lor\; b)\; \backslash lor\; c$
  Colin Howson (1997). 9780415133425, Routledge. ISBN 9780415133425
• Commutativity: $a\; \backslash lor\; b\; \backslash equiv\; b\; \backslash lor\; a$
• Distributivity: $(a\; \backslash land\; (b\; \backslash lor\; c))\; \backslash equiv\; ((a\; \backslash land\; b)\; \backslash lor\; (a\; \backslash land\; c))$

::$(a\; \backslash lor\; (b\; \backslash land\; c))\; \backslash equiv\; ((a\; \backslash lor\; b)\; \backslash land\; (a\; \backslash lor\; c))$

::$(a\; \backslash lor\; (b\; \backslash lor\; c))\; \backslash equiv\; ((a\; \backslash lor\; b)\; \backslash lor\; (a\; \backslash lor\; c))$

::$(a\; \backslash lor\; (b\; \backslash equiv\; c))\; \backslash equiv\; ((a\; \backslash lor\; b)\; \backslash equiv\; (a\; \backslash lor\; c))$

• Idempotency: $a\; \backslash lor\; a\; \backslash equiv\; a$
• Monotonicity: $(a\; \backslash rightarrow\; b)\; \backslash rightarrow\; ((c\; \backslash lor\; a)\; \backslash rightarrow\; (c\; \backslash lor\; b))$

::$(a\; \backslash rightarrow\; b)\; \backslash rightarrow\; ((a\; \backslash lor\; c)\; \backslash rightarrow\; (b\; \backslash lor\; c))$

• Truth-preserving: The interpretation under which all variables are assigned a truth value 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), the logical disjunction operator returns one of its operands: the first operand if it evaluates to a true value, and the second operand otherwise. This allows it to fulfill the role of the Elvis operator.

---

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: $x\backslash in\; A\backslash cup\; B\backslash Leftrightarrow\; (x\backslash in\; A)\backslash vee(x\backslash in\; B)$. Because of this, logical disjunction satisfies many of the same identities as set-theoretic union, such as associativity, commutativity, distributivity, and de Morgan's laws, identifying logical conjunction with set intersection, logical negation with set complement.

(2025). 9783662638651, Springer. ISBN 9783662638651

---

Natural language Disjunction in does not precisely match the interpretation of $\backslash lor$ in classical logic. Notably, classical disjunction is inclusive while natural language disjunction is often understood exclusively, as the following English example typically would be.

* 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 language soit... soit have been argued to be inherently exclusive, rendering ungrammaticality 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-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.

* You can have an apple or a pear.

:$\backslash rightsquigarrow$ You can have an apple and you can have a pear (but you cannot 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 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.

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 language and Maricopa, disjunction is marked using a verb suffix. For instance, in the Maricopa example below, disjunction is marked by the suffix šaa.

---

See also

• Affirming a disjunct
• Boolean algebra (logic)
• Boolean algebra topics
• Boolean domain
• Boolean function
• Boolean-valued 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 that does not necessitate mutual exclusiveness.

---

External links

• Eric W. Weisstein. "Disjunction." From MathWorld—A Wolfram Web Resource

Categories: Disjunction, Semantics, Formal Semantics

Post Comment