The modal operators usually associated with the notions of possibility and necessity are classically duals. This paper aims to defy that duality in a paraconsistent environment, namely in a Belnapian Hybrid logic where both propositional variables and accessibility relations are four-valued. Hybrid logic, which is an extension of Modal logic, incorporates extra machinery such as nominals – for uniquely naming states – and a satisfaction operator – so that the formula under its scope is evaluated in the state whose name the satisfaction operator indicates.

In classical Hybrid logic the semantics of negation, when it appears before compound formulas, is carried towards subformulas, meaning that eventual inconsistencies can be found at the level of nominals or propositional variables but appear unrelated to the accessibility relations. In this paper we allow inconsistencies in propositional variables and, by breaking the duality between modal operators, inconsistencies at the level of accessibility relations arise. We introduce a sound and complete tableau system and a decision procedure to check if a formula is a consequence of a set of formulas. Tableaux will be used to extract syntactic models for databases, which will then be compared using different inconsistency measures. We conclude with a discussion about bisimulation.

通常与可能性和必要性概念相关的模态运算符在经典意义上是对偶。本文旨在克服超常一致性环境中的对偶性，即命题变量和可及性关系均为四值的Belnapian混合逻辑。混合逻辑是模态逻辑的扩展，它结合了额外的机制，例如标称符号（用于唯一命名状态）和满意度运算符，以便在其范围内的公式在其名称由满意度运算符指示的状态下进行评估。

在经典的混合逻辑中，否定的语义在出现在复合公式之前时，会带到子公式，这意味着最终的不一致可以在名词或命题变量的级别上找到，但看起来与可及性关系无关。在本文中，我们允许命题变量不一致，并且通过打破模态运算符之间的对偶关系，在可访问性关系级别上会出现不一致。我们引入了完善而完整的表格系统和决策程序，以检查某个公式是否是一组公式的结果。Tableaux将用于提取数据库的句法模型，然后将使用不同的不一致度量进行比较。我们以关于双仿真的讨论作为结束。