The relationship between logics with sets of theorems including contradictions (“inconsistent logics”) and theories closed under such logics is investigated. It is noted that if we take “theories” to be defined in terms of deductive closure understood in a way somewhat different from the standard, Tarskian, one, inconsistent logics can have consistent theories. That is, we can find some sets of formulas the closure of which under some inconsistent logic need not contain any contradictions. We prove this in a general setting for a family of relevant connexive logics, extract the essential features of the proof in order to obtain a sufficient condition for the consistency of a theory in arbitrary logics, and finally consider some concrete examples of consistent mathematical theories in Abelian logic. The upshot is that on this way of understanding deductive closure, common to relevant logics, there is a rich and interesting kind of interaction between inconsistent logics and their theories. We argue that this suggests an important avenue for investigation of inconsistent logics, from both a technical and a philosophical perspective.

研究了包含矛盾（“不一致逻辑”）的定理集逻辑与在这种逻辑下封闭的理论之间的关系。值得注意的是，如果我们根据演绎闭合来定义“理论”，其理解方式与标准塔斯基安的理解方式有些不同，那么不一致的逻辑可以有一致的理论。也就是说，我们可以找到一些公式集，在某些不一致的逻辑下，其闭包不需要包含任何矛盾。我们在相关连接逻辑族的一般设置中证明了这一点，提取证明的基本特征以获得理论在任意逻辑中的一致性的充分条件，最后考虑一些一致的数学理论的具体例子阿贝尔逻辑。结果是，在这种理解演绎闭包（相关逻辑共有）的方式中，不一致的逻辑与其理论之间存在着丰富而有趣的相互作用。我们认为，这表明从技术和哲学角度研究不一致逻辑的重要途径。