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
v t e

Logical connectives
AND ${\displaystyle A\land B}$, ${\displaystyle A\cdot B}$, ${\displaystyle AB}$, ${\displaystyle A\&B}$, ${\displaystyle A\&\&B}$ equivalent ${\displaystyle A\equiv B}$, ${\displaystyle A\Leftrightarrow B}$, ${\displaystyle A\leftrightharpoons B}$ implies ${\displaystyle A\Rightarrow B}$, ${\displaystyle A\supset B}$, ${\displaystyle A\rightarrow B}$ NAND ${\displaystyle A{\overline {\land }}B}$, ${\displaystyle A\uparrow B}$, ${\displaystyle A\mid B}$, ${\displaystyle {\overline {A\cdot B}}}$ nonequivalent ${\displaystyle A\not \equiv B}$, ${\displaystyle A\not \Leftrightarrow B}$, ${\displaystyle A\nleftrightarrow B}$ NOR ${\displaystyle A{\overline {\lor }}B}$, ${\displaystyle A\downarrow B}$, ${\displaystyle {\overline {A+B}}}$ NOT ${\displaystyle \neg A}$, ${\displaystyle -A}$, ${\displaystyle {\overline {A}}}$, ${\displaystyle \sim A}$ OR ${\displaystyle A\lor B}$, ${\displaystyle A+B}$, ${\displaystyle A\mid B}$, ${\displaystyle A\parallel B}$ XNOR ${\displaystyle A\ {\text{XNOR}}\ B}$ XOR ${\displaystyle A{\underline {\lor }}B}$, ${\displaystyle A\oplus B}$ converse ${\displaystyle A\Leftarrow B}$, ${\displaystyle A\subset B}$, ${\displaystyle A\leftarrow B}$
Related concepts
Propositional calculus Predicate logic Boolean algebra Truth table Truth function Boolean function Functional completeness
Applications
Digital logic Programming languages Mathematical logic Philosophy of logic
Category
v t e

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]

An operand of a disjunction is a disjunct.^[3]

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).^[4]

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:^[5]

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

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

${\displaystyle A}$	${\displaystyle B}$	${\displaystyle A\lor B}$
F	F	F
F	T	T
T	F	T
T	T	T

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" ($$Zdroj:https://en.wikipedia.org?pojem=Logical_or
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