This paper concerns an expansion of first-order Belnap-Dunn logic named $\mathrm{BD}^{\supset,\mathsf{F}}$. Its connectives and quantifiers are all familiar from classical logic and its logical consequence relation is closely connected to the one of classical logic. Results that convey this close connection are established. Classical laws of logical equivalence are used to distinguish the four-valued logic $\mathrm{BD}^{\supset,\mathsf{F}}$ from all other four-valued logics with the same connectives and quantifiers whose logical consequence relation is as closely connected to the logical consequence relation of classical logic. It is shown that several interesting non-classical connectives added to Belnap-Dunn logic in its studied expansions are definable in $\mathrm{BD}^{\supset,\mathsf{F}}$. It is also established that $\mathrm{BD}^{\supset,\mathsf{F}}$ is both paraconsistent and paracomplete. A sequent calculus proof system that is sound and complete with respect to the logical consequence relation of $\mathrm{BD}^{\supset,\mathsf{F}}$ is presented.

翻译：本文研究一阶Belnap-Dunn逻辑的一个扩张，命名为$\mathrm{BD}^{\supset,\mathsf{F}}$。该逻辑的联结词与量词均源于经典逻辑，其逻辑后承关系与经典逻辑密切相关。文中建立了体现这种紧密联系的相关结论。通过经典逻辑等价律，我们区分了四值逻辑$\mathrm{BD}^{\supset,\mathsf{F}}$与其他具有相同联结词与量词、且逻辑后承关系与经典逻辑同样密切关联的四值逻辑。研究表明，在Belnap-Dunn逻辑已有扩张中引入的若干非经典联结词，均可定义于$\mathrm{BD}^{\supset,\mathsf{F}}$。同时证明$\mathrm{BD}^{\supset,\mathsf{F}}$兼具亚协调性与亚完备性。最后给出一个与$\mathrm{BD}^{\supset,\mathsf{F}}$逻辑后承关系相匹配的、具有可靠性和完备性的矢列演算证明系统。