This paper concerns an expansion of first-order Belnap-Dunn logic which is called $\mathrm{BD}^{\supset,\mathsf{F}}$. Its connectives and quantifiers are all familiar from classical logic and its logical consequence relation is very closely connected to the one of classical logic. Results that convey this close connection are established. Fifteen classical laws of logical equivalence are used to distinguish $\mathrm{BD}^{\supset,\mathsf{F}}$ 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 expansions that have been studied earlier are definable in $\mathrm{BD}^{\supset,\mathsf{F}}$. It is also established that $\mathrm{BD}^{\supset,\mathsf{F}}$ is both paraconsistent and paracomplete. Moreover, 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：本文关注一阶Belnap-Dunn逻辑的一种扩张，称为$\mathrm{BD}^{\supset,\mathsf{F}}$。其联结词与量词均取自经典逻辑，且其逻辑后承关系与经典逻辑的逻辑后承关系紧密相连。文中确立了体现这种紧密联系的相关结论。通过十五个经典逻辑等价定律，可将$\mathrm{BD}^{\supset,\mathsf{F}}$与所有其他具有相同联结词和量词、且其逻辑后承关系与经典逻辑的逻辑后承关系同样紧密相连的四值逻辑区分开来。研究表明，在Belnap-Dunn逻辑的扩张中已被研究过的若干有趣的非经典联结词，可在$\mathrm{BD}^{\supset,\mathsf{F}}$中定义。此外，还确立了$\mathrm{BD}^{\supset,\mathsf{F}}$同时具有悖论包容性和悖论完全性的性质。最后，本文给出了一个相对于$\mathrm{BD}^{\supset,\mathsf{F}}$逻辑后承关系既可靠又完备的矢列演算证明系统。