Paper detail

Filter pairs and natural extensions of logics

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.