disjunktio

Disjunktio, also known as disjunction, is a logical connective that represents the operation of "or." In propositional logic, A disjuncts B means that at least one of the propositions A or B is true. The standard symbol is the logical OR, commonly written as A ∨ B, and it is read as “A or B.” In classical logic, disjunction is inclusive, meaning that A ∨ B is true if A is true, or B is true, or both are true. The term inclusive or is used to distinguish it from the exclusive or (XOR), which is true when exactly one of A or B is true.

The truth-functional definition of A ∨ B is often summarized by a truth table: if A and B are both true, A ∨ B is true; if A is true and B is false, A ∨ B is true; if A is false and B is true, A ∨ B is true; if both are false, A ∨ B is false. The disjunction is commutative (A ∨ B ≡ B ∨ A) and associative ((A ∨ B) ∨ C ≡ A ∨ (B ∨ C)). Its identity element is false, since false ∨ A ≡ A, and true is an absorbing element since true ∨ A ≡ true for any A. De Morgan’s laws connect negation and disjunction: ¬(A ∨ B) ≡ (¬A) ∧ (¬B).

Beyond propositional logic, disjunction corresponds to the union operation in set theory (P ∪ Q) and to the logical OR operator in programming languages (often short-circuit evaluated). In proofs, disjunction is used in introduction and elimination rules to handle alternatives.