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.

翻译：本文关注一阶Belnap-Dunn逻辑的一种扩张，该逻辑的联结词和量词均源于经典逻辑。文中定义了该逻辑的语言和逻辑后承关系，给出了所定义逻辑的证明系统，并建立了该证明系统的可靠性与完备性。通过所给证明系统与具有相同语言的经典逻辑版本的可靠且完备证明系统之间的细微差异，阐明了所定义逻辑的逻辑后承关系与相同语言的经典逻辑版本之间的紧密关联。此外，文中给出了十五条经典逻辑等价律，这些定律使所定义逻辑的逻辑等价关系与许多初看密切相关的逻辑的逻辑等价关系区分开来。