Normal extensions of KTB of codimension 3

TL;DR

This paper constructs an uncountable family of graphs to demonstrate that there are continuum many normal extensions of the logic KTB of codimension 3, using algebraic methods for the proof.

Contribution

It generalizes finite examples to an uncountable family, showing the vast diversity of normal extensions of KTB at codimension 3.

Findings

Existence of uncountably many normal extensions of KTB of codimension 3

Construction of a continuum of such extensions using algebraic methods

Demonstration that algebraic methods are more effective than frame-theoretic ones in this context

Abstract

It is known that in the lattice of normal extensions of the logic KTB there are unique logics of codimensions 1 and 2, namely, the logic of a single reflexive point, and the logic of the total relation on two points. A natural question arises about the cardinality of the set of normal extensions of KTB of codimension 3. Generalising two finite examples found by a computer search, we construct an uncountable family of (countable) graphs, and prove that certain frames based on these produce a continuum of normal extensions of KTB of codimension 3. We use algebraic methods, which in this case turn out to be better suited to the task than frame-theoretic ones.