Multiple-conclusion Hilbert-style systems allow us to finitely axiomatize every logic defined by a finite matrix. Having obtained such axiomatizations for Paraconsistent Weak Kleene and Bochvar-Kleene logics, we modify them by replacing the multiple-conclusion rules with carefully selected single-conclusion ones. In this way we manage to introduce the first finite Hilbert-style single-conclusion axiomatizations for these logics.

翻译：多结论希尔伯特风格系统使我们能够对每个由有限矩阵定义的逻辑进行有限公理化。在获得帕拉一致弱克莱恩逻辑和博赫瓦尔-克莱恩逻辑的此类公理化后，我们通过用精心选择的单结论规则替换多结论规则对其进行修改。通过这种方式，我们成功地为这些逻辑引入了首个有限的单结论希尔伯特风格公理化系统。