On distance logics of Euclidean spaces

TL;DR

This paper introduces and analyzes distance-based modal logics derived from Euclidean spaces, revealing their expressive power, distinctions across dimensions, and properties like non-finite axiomatizability.

Contribution

It defines new geometric modal logics from Euclidean spaces and proves their distinctness, expressive capabilities, and logical properties such as non-finite axiomatizability.

Findings

Farness, nearness, and constant distance logics are all distinct for different Euclidean spaces.

Farness and nearness logics of the real numbers strictly contain those of the rationals.

The farness logic of the reals is not finitely axiomatizable and lacks the finite model property.

Abstract

We consider logics derived from Euclidean spaces $\mathbb{R}^n$. Each Euclidean space carries relations consisting of those pairs that are, respectively, distance more than 1 apart, distance less than 1 apart, and distance 1 apart. Each relation gives a uni-modal logic of $\mathbb{R}^n$ called the farness, nearness, and constant distance logics, respectively. These modalities are expressive enough to capture various aspects of the geometry of $\mathbb{R}^n$ related to bodies of constant width and packing problems. This allows us to show that the farness logics of the spaces $\mathbb{R}^n$ are all distinct, as are the nearness logics, and the constant distance logics. The farness and nearness logics of $\mathbb{R}$ are shown to strictly contain those of $\mathbb{Q}$, while their constant distance logics agree. It is shown that the farness logic of the reals is not finitely axiomatizable and does not have the finite model property.