Filter pairs and natural extensions of logics

Abstract

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 \(\kappa \), where \(\kappa \) is a regular cardinal. The corresponding new notion is called \(\kappa \)-filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different \(\kappa \)-filter pairs give rise to a fixed logic of cardinality \(\kappa \). 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 \(\kappa \). Along the way we use \(\kappa \)-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.