Disjunktion

Disjunktion, in logic commonly known as logical disjunction, is a binary connective that represents the idea that at least one of two propositions is true. Its symbol is usually A ∨ B. The value of A ∨ B is true whenever A is true, or B is true, or both are true.

Truth-functional semantics: A ∨ B is true in all cases except when both A and B are false. The standard truth table shows: A B → A ∨ B; false false → false; false true → true; true false → true; true true → true. In standard logic, disjunction is inclusive, meaning both operands can be true simultaneously.

Variants: An exclusive or (XOR), true only if exactly one of A or B is true, is a different connective often denoted ⊕ or XOR. Inclusive disjunction and exclusive disjunction are related but yield different truth values when both operands are true.

Algebraic properties: Disjunction is commutative (A ∨ B = B ∨ A) and associative (A ∨ (B ∨ C) = (A ∨ B) ∨ C). The neutral element is False: A ∨ False = A, while True makes the expression True regardless of A: A ∨ True = True. It distributes over conjunction: A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C). De Morgan’s laws connect negation with disjunction: ¬(A ∨ B) = ¬A ∧ ¬B.

Applications: Disjunction is fundamental in mathematics, computer science, and language semantics. It forms larger statements, defines truth conditions, and drives control flow in programming. In digital logic, ∨ corresponds to OR gates; in SQL and many programming languages, the OR operator combines conditions, often with short-circuit evaluation.

Origin: The concept is central to Boolean algebra developed by George Boole, foundational for propositional logic and digital computation.