We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety.

我们研究四值 Belnap-Dunn 逻辑的扩展格，通过类比超直觉逻辑称为超 Belnap 逻辑。我们通过将这个格子分成几个子区间来描述这个格子的全局结构，并证明了一些新的超 Belnap 逻辑完备性定理。为此目的的关键技术工具将是所谓的反公理化（或爆炸性）部分运算符。由于与图论的联系，Belnap-Dunn 逻辑的反公理化（或爆炸性）扩展变得特别有趣：Belnap-Dunn 逻辑的有限反公理化扩展的格同构于同态序中的翻转格有限图（允许循环）。特别是，有一个有限的超 Belnap 逻辑的连续统一体。而且，在这种同构的帮助下，可以构造一个非有限的超 Belnap 逻辑。作为代数推论，我们得到 De Morgan 代数的反簇连续统的存在性以及 De Morgan 代数的预簇的存在性，它不是准簇。