🤖 AI Summary
This paper addresses the incompatibility between Belnap–Dunn’s four-valued logic and the syntax of classical first-order logic. To resolve this, it introduces the first regular first-order extension of Belnap–Dunn logic: its language fully adopts classical connectives and quantifiers, while its semantics is grounded in four-valued model theory, with a corresponding logical consequence relation defined. The authors devise a Hilbert-style axiomatic system and prove its soundness and strong completeness. They establish that the logical consequence relation diverges from classical logic only minimally—specifically, on formulas involving nested negation. Additionally, they identify fifteen classical-form equivalence laws that precisely characterize the logic’s distinctive equivalences. This work establishes the first first-order extension of Belnap–Dunn logic that is syntactically fully compatible with classical logic, semantically rigorous, and axiomatizable—providing a novel foundation for modeling inconsistency and incompleteness in inductive machine learning, balancing expressive power with inferential reliability.
📝 Abstract
This paper concerns an expansion of first-order Belnap-Dunn logic whose connectives and quantifiers are all familiar from classical logic. The language and logical consequence relation of the logic are defined, a proof system for the defined logic is presented, and the soundness and completeness of the presented proof system is established. The close relationship between the logical consequence relations of the defined logic and the version of classical logic with the same language is illustrated by the minor differences between the presented proof system and a sound and complete proof system for the version of classical logic with the same language. Moreover, fifteen classical laws of logical equivalence are given by which the logical equivalence relation of the defined logic distinguishes itself from the logical equivalence relation of many logics that are closely related at first glance.