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：多结论希尔伯特式系统允许我们有限公理化每一个由有限矩阵定义的逻辑。在获得悖论弱克莱恩逻辑和博赫瓦-克莱恩逻辑的此类公理化后，我们通过用精心选取的单结论规则替换多结论规则来修改它们。通过这种方式，我们成功引入了这些逻辑的首个有限希尔伯特式单结论公理化系统。