Spatial logics are modal logics whose modalities are interpreted using topological concepts of neighbourhood and connectivity. Recently, these logics have been extended to (pre)closure spaces, a generalization of topological spaces covering also the notion of neighbourhood in discrete structures. In this paper we introduce an abstract theoretical framework for the systematic investigation of the logical aspects of closure spaces. To this end we define the categorical notion of closure (hyper)doctrine, which are doctrines endowed with inflationary operators (and subject to suitable conditions). The generality and effectiveness of this notion is demonstrated by many examples arising naturally from topological spaces, fuzzy sets, algebraic structures, coalgebras, and covering at once also known cases such as Kripke frames and probabilistic frames (i.e., Markov chains). In order to model also surroundedness, closure hyperdoctrines are then endowed with paths; this construction allows us to cover all the logical constructs of the Spatial Logic for Closure Spaces. By leveraging general categorical constructions, we provide a first axiomatisation and sound and complete semantics for propositional/regular/first order logics for closure operators. Therefore, closure hyperdoctrines are useful both for refining and improving the theory of existing spatial logics, but especially for the definition of new spatial logics for various applications.

中文翻译：

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关闭超教义，带路径

空间逻辑是模态逻辑，其模态使用邻域和连通性的拓扑概念进行解释。最近，这些逻辑已经扩展到（预）闭合空间，拓扑空间的概括也涵盖了离散结构中邻域的概念。在本文中，我们介绍了一个抽象的理论框架，用于系统地研究封闭空间的逻辑方面。为此，我们定义了闭包（超）学说的分类概念，这些学说具有膨胀算子（并受适当条件的约束）。这个概念的普遍性和有效性通过从拓扑空间、模糊集、代数结构、余代数自然产生的许多例子来证明，并同时涵盖了已知的情况，如 Kripke 框架和概率框架（即。例如，马尔可夫链）。为了模拟环绕性，封闭超教义被赋予了路径；这种构造使我们能够涵盖闭包空间的空间逻辑的所有逻辑构造。通过利用一般分类构造，我们为闭包运算符的命题/正则/一阶逻辑提供了第一公理化和合理且完整的语义。因此，闭包超学说对于提炼和改进现有空间逻辑的理论非常有用，尤其是对于为各种应用定义新的空间逻辑。通过利用一般分类构造，我们为闭包运算符的命题/正则/一阶逻辑提供了第一公理化和合理且完整的语义。因此，闭包超学说对于提炼和改进现有空间逻辑的理论非常有用，尤其是对于为各种应用定义新的空间逻辑。通过利用一般的分类构造，我们为闭包运算符的命题/正则/一阶逻辑提供了第一公理化和合理且完整的语义。因此，闭包超学说对于提炼和改进现有空间逻辑的理论非常有用，尤其是对于为各种应用定义新的空间逻辑。