论文标题
过滤对和逻辑的自然扩展
Filter pairs and natural extensions of logics
论文作者
论文摘要
我们调整了用于创建和分析Finality Logics的Finality Filter对的概念，以便我们可以处理基数$κ$的逻辑，其中$κ$是常规的红衣主教。相应的新概念称为$κ$ filter对。过滤器对可以看作是逻辑的呈现，我们询问哪些$κ$ - 滤波器对产生了哪些固定的基数逻辑$κ$。为了使问题定义明确，我们局限于过滤器对的子集合，并通过一组基数变量$κ$来建立从该集合到该逻辑的自然扩展的一组。 在此过程中，我们使用$κ$ - 滤波器对为给定的逻辑构建自然扩展，确定该结构与文献中提出的其他几个之间的关系，并表明自然扩展的集合形成了完整的晶格。 在可选的部分中，我们介绍并激发了通用过滤器对的概念。
We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality $κ$, where $κ$ is a regular cardinal. The corresponding new notion is called $κ$-filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different $κ$-filter pairs give rise to a fixed logic of cardinality $κ$. To make the question well-defined we restrict to a subcollection of filter pairs and establish a bijection from that collection to the set of natural extensions of that logic by a set of variables of cardinality $κ$. Along the way we use $κ$-filter pairs to construct natural extensions for a given logic, work out the relationships between this construction and several others proposed in the literature, and show that the collection of natural extensions forms a complete lattice. In an optional section we introduce and motivate the concept of a general filter pair.
