BelnapDunn

Belnap-Dunn logic, also known as Belnap–Dunn four-valued logic or First-Degree Entailment (FDE), is a non-classical logic introduced by Nuel Belnap and Joseph Dunn in 1973 to formalize reasoning with incomplete and potentially conflicting information. It treats truth as a matter of degree or state rather than a single true/false dichotomy.

The logic uses four truth values: true (t), false (f), both true and false (b), and neither true nor false (n). This fourfold valuation allows a statement to be true, false, both, or undetermined, reflecting partial knowledge or inconsistency about information. In standard presentations, a set of designated values is used to define entailment; typically, t and b are designated, while f and n are not, which supports a paraconsistent and paracomplete reasoning stance.

Belnap-Dunn logic is paraconsistent, meaning it does not explode—i.e., from a contradiction (A and not A) one cannot derive every proposition. It is also paracomplete, allowing for truth-value gaps where information is incomplete. The standard connectives—negation, conjunction, disjunction, and implication—are defined to operate over the four values in a way that preserves these properties. Negation generally swaps t and f while leaving b and n in a symmetric or fixed role; conjunction and disjunction extend from the classical case but respect the four-valued semantics.

Historically, Belnap-Dunn logic provided a foundational four-valued framework for reasoning with conflicting or partial data and has influenced the development of paraconsistent logics and four-valued semantic theories. It is often discussed in relation to the First-Degree Entailment (FDE) family and related systems. Applications appear in knowledge representation, databases, information integration, and reasoning under uncertainty, where data may be incomplete or contradictory.