Hybrid logic for strict betweenness

TL;DR

This paper explores the modal properties of the strict betweenness relation in geometry, demonstrating its non-definability in certain logical frameworks and proposing hybrid logic systems for dense linear frames.

Contribution

It introduces two hybrid logic systems tailored for dense linear betweenness frames, one complete for dense frames without endpoints and the other for Dedekind complete frames.

Findings

Strict betweenness is non-definable in basic similarity logic.

Two hybrid logic systems are developed for different classes of betweenness frames.

One system is complete for dense linear frames without endpoints.

Abstract

The paper is devoted to modal properties of the ternary strict betweenness relation as used in the development of various systems of geometry. We show that such a relation is non-definable in a basic similarity type with a binary operator of possibility, and we put forward two systems of hybrid logic, one of them complete with respect to the class of dense linear betweenness frames without endpoints, and the other with respect to its subclass composed of Dedekind complete frames.