Filter pairs and natural extensions of logics

TL;DR

This paper extends the concept of filter pairs to handle logics of arbitrary regular cardinality, establishing a correspondence with natural extensions and analyzing their lattice structure.

Contribution

It introduces the notion of $oldsymbol{ ext{kappa}}$-filter pairs for logics of size $oldsymbol{ ext{kappa}}$, connecting them to natural extensions and exploring their lattice properties.

Findings

$oldsymbol{ ext{kappa}}$-filter pairs characterize logics of size $oldsymbol{ ext{kappa}}$

A bijection between filter pairs and natural extensions is established

The set of natural extensions forms a complete lattice

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.