A dichotomy for some elementarily generated modal logics

TL;DR

This paper investigates a class of normal modal logics defined by specific first-order formulas, revealing a dichotomy where key properties either all hold or all fail together for these logics.

Contribution

It establishes a comprehensive dichotomy for properties of elementary classes defined by certain first-order formulas in modal logic.

Findings

Properties like finite axiomatisability and elementarity are linked in a dichotomous manner.

Many properties either all hold or all fail simultaneously for these modal logics.

The results unify various criteria for modal definability and axiomatisability.

Abstract

In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form $\forall x_0 \exists x_1 \dots \exists x_n \bigwedge x_i R_\lambda x_j$. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.