Disjunction

Disjunction is a truth-functional operator in logic which is equivalent to the word “or”, or more specifically “and/or”.

If either P is known or Q is known, we may say P or Q, or formally:

P ∨ Q

This may be read as “P or Q” or “it is the case that P, Q, or both”.

In symbolic logic, the disjunction symbol ( ∨ ) is used to symbolize a disjunction. Sometimes, the capitalized word OR is used.

Truth values

The following table illustrates the possible truth values of P ∨ Q, given each possible valuation of its terms, P and Q. Note that P ∨ Q is true if either P or Q is true, and false only when both are false.

Inclusive vs. exclusive disjunctions

The ∨ symbol is used to express an inclusive disjunction, because it includes the possibility that the two connected propositions or formulae are both true. This means that if P ∨ Q, then either P, Q, or both are true.

In some cases, an exclusive disjunction is needed to convey instances where P and Q are mutually exclusive, but one is true — that is, P or Q, but not both. Some systems of logic introduce another symbol, usually ⊕, to handle an exclusive disjunction (such as P ⊕ Q). In systems that do not use such as symbol, an exclusive disjunction may be expressed as [(P ∨ Q) ∧ ¬(P ∧ Q)], which reads as “P or Q, but not both”.