In order to develop efficient tools for automated reasoning with inconsistency (theorem provers), eventually making Logics of Formal inconsistency (LFI) a more appealing formalism for reasoning under uncertainty, it is important to develop the proof theory of the first-order versions of such LFIs. Here, we intend to make a first step in this direction. On the other hand, the logic Ciore was developed to provide new logical systems in the study of inconsistent databases from the point of view of LFIs. An interesting fact about Ciore is that it has a strong consistency operator, that is, a consistency operator which (forward/backward) propagates inconsistency. Also, it turns out to be an algebraizable logic (in the sense of Blok and Pigozzi) that can be characterized by means of a 3-valued logical matrix. Recently, a first-order version of Ciore, namely QCiore, was defined preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. Besides, some important model-theoretic results were obtained for this logic. In this paper we study some proof–theoretic aspects of both Ciore and QCiore respectively. In first place, we introduce a two-sided sequent system for Ciore. Later, we prove that this system enjoys the cut-elimination property and apply it to derive some interesting properties. Later, we extend the above-mentioned system to first-order languages and prove completeness and cut-elimination property using the well-known Shütte’s technique.

为了开发用于不一致性自动推理（定理证明器）的有效工具，最终使形式不一致性逻辑（LFI）成为不确定性下推理的更具吸引力的形式主义，开发此类的一阶版本的证明理论非常重要LFI。在这里，我们打算朝这个方向迈出第一步。另一方面，逻辑Ciore的开发是为了从LFI 的角度研究不一致数据库提供新的逻辑系统。关于Ciore的一个有趣的事实 是，它有一个强一致性算子，即（向前/向后）传播不一致的一致性算子。而且，它被证明是一种可代数逻辑（在 Blok 和 Pigozzi 的意义上），可以通过 3 值逻辑矩阵来表征。最近，定义了Ciore的一阶版本，即QCiore ，保留了Ciore的精神，即没有在量词之间引入意外的关系。此外，该逻辑还获得了一些重要的模型理论结果。在本文中，我们分别研究了Ciore 和QCIore的一些证明理论方面 。首先，我们为Ciore引入双边序列系统。后来，我们证明了该系统具有割消除性质，并应用它导出了一些有趣的性质。后来，我们将上述系统扩展到一阶语言，并使用著名的Shütte技术证明了完整性和割消性。