Trivalued

Trivalued is an adjective used to describe something that possesses three distinct values. In logic and computer science, trivalued commonly refers to three-valued logics, which extend classical two-valued (true/false) logic by introducing a third value, often interpreted as indeterminate, unknown, or both true and false.

Two influential formulations are Kleene's three-valued logic (K3) and Łukasiewicz's three-valued logic (Ł3). In K3, the truth values are true, false, and undefined. Logical connectives have truth tables designed to reflect the intuition that indeterminacy can propagate. For example, in conjunction, false dominates: F ∧ x = F; true with an indeterminate yields indeterminate: T ∧ U = U; while negation fixes T and F and leaves U unchanged (¬T = F, ¬F = T, ¬U = U). In disjunction, true dominates: T ∨ x = T; F ∨ U = U; while other combinations follow the same principle of indeterminacy propagation.

In Ł3, truth values are often represented as 0, 1/2, and 1, with negation ¬x = 1 − x, conjunction x ∧ y = min(x, y), disjunction x ∨ y = max(x, y), and implication x → y = min(1, 1 − x + y). This arithmetic-based approach yields a distinct logical calculus from Kleene's.

Other approaches, such as Bochvar’s internal three-valued logic, assign a third value to denote meaningless or undecidable propositions, with this value propagating through formulas.

Applications span theoretical investigations into semantics of partial information, modeling uncertainty, and database systems that treat missing data via a NULL value that behaves like a third truth value.

See also three-valued logic, multi-valued logic, Kleene logic, Łukasiewicz logic.