The 3-valued paraconsistent logic Ciore was developed by Carnielli, Marcos and de Amo under the name LFI2, in the study of inconsistent databases from the point of view of logics of formal inconsistency (LFIs). They also considered a first-order version of Ciore called LFI2*. The logic Ciore enjoys extreme features concerning propagation and retropropagation of the consistency operator: a formula is consistent if and only if some of its subformulas is consistent. In addition, Ciore is algebraizable in the sense of Blok and Pigozzi. On the other hand, the logic LFI2* satisfies a somewhat counter-intuitive property: the universal and the existential quantifier are inter-definable by means of the paraconsistent negation, as it happens in classical first-order logic with respect to the classical negation. This feature seems to be unnatural, given that both quantifiers have the classical meaning in LFI2*, and that this logic does not satisfy the De Morgan laws with respect to its paraconsistent negation. The first goal of the present article is to introduce a first-order version of Ciore (which we call QCiore) preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. The second goal of the article is to adapt to QCiore the partial structures semantics for the first-order paraconsistent logic LPT1 introduced by Coniglio and Silvestrini, which generalizes the semantic notion of quasi-truth considered by Mikeberg, da Costa and Chuaqui. Finally, some important results of classical Model Theory are obtained for this logic, such as Robinson’s joint consistency theorem, amalgamation and interpolation. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order LFIs.

中文翻译：

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三值次协一阶逻辑QCIORE的一些模型-理论结果

三值次一致性逻辑乔雷由 Carnielli、Marcos 和 de Amo 以名称开发LFI2, 在研究不一致的数据库时形式不一致的逻辑(LFIs)。他们还考虑了一阶版本的乔雷叫LFI2*。逻辑乔雷具有关于一致性算子的传播和逆向传播的极端特征：一个公式是一致的当且仅当它的一些子公式是一致的。此外，乔雷在 Blok 和 Pigozzi 的意义上是可代数的。另一方面，逻辑LFI2* 满足了一个有点违反直觉的性质：全称量词和存在量词可以通过副一致的否定相互定义，就像经典一阶逻辑中关于经典否定的情况一样。鉴于这两个量词在LFI2*，并且这个逻辑不满足德摩根定律关于它的平行否定。本文的第一个目标是介绍一阶版本的乔雷（我们称之为QCiore) 保持精神乔雷，也就是说，不会在量词之间引入意想不到的关系。文章的第二个目标是适应QCiore一阶副一致逻辑的部分结构语义LPT1由 Coniglio 和 Silvestrini 引入，它概括了准真理由 Mikeberg、da Costa 和 Chuaqui 考虑。最后，针对这一逻辑，得到了经典模型理论的一些重要结果，如鲁宾逊联合一致性定理、合并和插值。虽然我们专注于QCiore，这个框架可以适应其他三值一阶LFIs。