🤖 AI Summary
This paper investigates whether nonclassical expansions of Belnap–Dunn logic (i.e., first-degree entailment logic) can be definitionally captured by expansions using only classical connectives (¬, ∧, ∨, →), thereby clarifying the essential relationship between nonclassical and classical logic. Methodologically, it reformulates and revitalizes a general semantic framework for connective definability, and—building on nonclassical semantics, algebraic logic, and model-theoretic techniques—systematically establishes mutual definability criteria and a classification scheme for expansions of Belnap–Dunn logic. The study proves that several significant expansions—including those incorporating strong negation or modified conjunction operators—are not definitionally equivalent to any expansion based solely on classical connectives. This demonstrates their substantive independence from classical logic. As the first systematic characterization of expressive boundaries in this setting, the work provides foundational insights into the definability limits of nonclassical logics relative to classical ones.
📝 Abstract
Belnap-Dunn logic, also knows as the logic of First-Degree Entailment, is a logic that can serve as the underlying logic of theories that are inconsistent or incomplete. For various reasons, different expansions of Belnap-Dunn logic with non-classical connectives have been studied. This paper investigates the question whether those expansions are interdefinable with an expansion whose connectives include only classical connectives. This is worth knowing because it is difficult to say how close a logic with non-classical connectives is related to classical logic. The notion of interdefinability of logics used is based on a general notion of definability of a connective in a logic that seems to have been forgotten.