In logic, logical disjunction is the join in the poset of truth values.
Assuming that (as in classical logic) the only truth values are true () and false (), then the disjunction of the truth values and may be defined by a truth table:
That is, is true if and only if at least one of and is true. Disjunction also exists in nearly every non-classical logic.
More generally, if and are any two relations on the same domain, then we define their disjunction pointwise, thinking of a relation as a function to truth values. If instead we think of a relation as a subset of its domain, then disjunction becomes union.
Disjunction as defined above is sometimes called inclusive disjunction to distinguish it from exclusive disjunction, where exactly one of and must be true.
In the context of substructural logics such as linear logic, it is also called additive disjunction to disambiguate it from the multiplicative disjunction.
Disjunction is de Morgan dual to conjunction.
Like any join, disjunction is an associative operation, so we can take the disjunction of any finite positive whole number of truth values; the disjunction is true if and only if at least one of the various truth values is true. Disjunction also has an identity element, which is the false truth value. Some logics allow a notion of infinitary disjunction. Indexed disjunction is existential quantification.
Rules of inference
The rules of inference? for disjunction in sequent calculus are dual to those for conjunction:
Equivalently, we can use the following rules with weakened contexts:
The rules above are written so as to remain valid in logics without the exchange rule. In linear logic, the first batch of sequent rules apply to additive disjunction (interpret in these rules as ), while the second batch of rules apply to multiplicative disjunction (interpret in those rules as ).
The natural deduction rules for disjunction are a little more complicated than those for conjunction: