fourvalued

Four-valued logic is a family of non-classical logics that extend classical true/false reasoning by incorporating two additional truth values to capture information that is incomplete or inconsistent. The standard scheme uses four truth values: true (T), false (F), both true and false (B), and neither true nor false (N). This allows logical reasoning to proceed in the presence of conflicting data or incomplete knowledge.

Semantics for four-valued logics are often built on bilattices, providing two independent orderings: an information order and a truth order. Operators such as conjunction, disjunction and negation are defined to respect these structures. A typical negation swaps T and F while leaving B and N fixed (not T = F, not F = T, not B = B, not N = N). Conjunction and disjunction are interpreted as the meet and join with respect to the information lattice, producing results that reflect both truth and information content. The resulting logics are usually paraconsistent (they do not explode in the face of a contradiction) and can handle both inconsistent and incomplete information without trivialization.

Belnap's four-valued logic, introduced by Nuel Belnap in 1977, is a canonical example. The framework has influenced database theory, knowledge representation, and logic programming, where data may be incomplete (N) or conflicting (B). Four-valued logics also serve as foundations for more elaborate paraconsistent systems and have been studied in various forms, including alternative truth operators and extensions to modal or temporal settings.