In this paper, we introduce a syntactic framework for analyzing and handling inconsistencies in propositional bases. Our approach focuses on examining the relationships between variable occurrences within conflicts. We propose two dual concepts: Minimal Inconsistency Relation (MIR) and Maximal Consistency Relation (MCR). Each MIR is a minimal equivalence relation on variable occurrences that results in inconsistency, while each MCR is a maximal equivalence relation designed to prevent inconsistency. Notably, MIRs capture conflicts overlooked by minimal inconsistent subsets. Using MCRs, we develop a series of non-explosive inference relations. The main strategy involves restoring consistency by modifying the propositional base according to each MCR, followed by employing the classical inference relation to derive conclusions. Additionally, we propose an unusual semantics that assigns truth values to variable occurrences instead of the variables themselves. The associated inference relations are established through Boolean interpretations compatible with the occurrence-based models.

翻译：本文提出了一种用于分析和处理命题基中不一致性的语法框架。我们的方法侧重于考察冲突中变量出现之间的关系。我们提出了两个对偶概念：最小不一致关系（MIR）和最大一致关系（MCR）。每个MIR是导致不一致性的变量出现上的最小等价关系，而每个MCR则是旨在防止不一致性的最大等价关系。值得注意的是，MIRs捕捉了最小不一致子集所忽略的冲突。利用MCRs，我们发展了一系列非爆炸性推理关系。主要策略包括根据每个MCR修改命题基以恢复一致性，然后运用经典推理关系来推导结论。此外，我们提出了一种不寻常的语义，它将真值分配给变量出现而非变量本身。相关的推理关系是通过与基于出现的模型相容的布尔解释建立的。