Logical disjunction

The logical disjunction , or nonexclusive disjunction of two events represents the fact that at least one of these two events occurs (one, or the other, or both).

In the Logical language or mathematical and in the technical fields which employ it, it results in the OR logical , a logical operator in the Calcul of the proposals. The proposal obtained by connecting two proposals by this operator is also called their disjunction, or them logical sum . The disjunction of two proposals P and Q is true when one of the proposals is true, and is false when both are simultaneously false. Disjunction is written:

P ∨ Q

and is read

“ P or Q ”

The symbol “∨” is called connector of disjunction.

The Truth table of a disjunction is given by the following table

Let us notice what, in the language running, the Coordinating conjuction “or” is employed in the direction “one or the other, but not both”, for example when we ask “will take you coffee or the? ”. In this case “or” an alternative indicates, and has the same direction as “or”. In logic that is called exclusive disjunction or the “Or exclusive”.

Formally, it “or” logic between two proposals is also true when the two proposals are true; thus it “or” is called also inclusive disjunction. This is returned better in the language running by the expression “and/or”.

Note: Boole, by analogy narrow with ordinary mathematics, imposed in the definition of X + there , the condition of mutual exclusion of X and there . William Jevons, and practically all the logicians in mathematics which succeeded to him, recommended for various reasons, the use of a definition of the logical sum not making obligatory mutual exclusion.

Disjunction that we described is a binary Operator, which means that it combines two proposals in only one. However, we can connect disjunctions, by considering for example has ∨ B ∨ C , which is by definition one or the other of the two logically equivalent proposals ( has ∨ B ) ∨ C or has ∨ ( B ∨ C ). This proposal is true when one of the proposals has , B , or C is true. The sequence of the conjunctions is made possible thanks to the Associativité of the ∨. The operator is also Commutatif; has ∨ B is equivalent to B ∨ has .

Let us give some properties of the conjunction:
Are P , Q and R three proposals.

( P ∨ P ) ⇔ P idempotence of “or”
( P ∨ Q ) ⇔ ( Q ∨ P ) commutation of “or”
(( P ∨ Q ) ∨ R ) ⇔ ( P ∨ ( Q ∨ R )) associativeness of “or”
¬ ( P ∨ Q ) ⇔ ((¬ P ) ∧ (¬ Q )) the negation of a disjunction is the conjunction of the negations
¬ ( P ∧ Q ) ⇔ ((¬ P ) ∨ (¬ Q )) the negation of a conjunction is the disjunction of the negations
( P ∨ ( Q ∧ R )) ⇔ (( P ∨ Q ) ∧ ( P ∨ R )) distributivity of “or” compared to “and”
( P ∧ ( Q ∨ R )) ⇔ (( P ∧ Q ) ∨ ( P ∧ R )) distributivity of “and” compared to “or”

The corresponding concept in set theory is the meeting.

See too

logical Conjunction
existential Quantifier

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Logical disjunction

See too

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