On modal logics of model-theoretic relations

TL;DR

This paper explores the modal logic frameworks based on model-theoretic relations, analyzing how these logics depend on language, their completeness, and expressibility, with specific results on submodel and quotient relations.

Contribution

It introduces a systematic study of modal logics derived from model-theoretic relations, including completeness and expressibility results, and proves a downward Löwenheim–Skolem theorem for expanded languages.

Findings

Calculated modal theories for submodel and quotient relations.

Established Kripke completeness conditions.

Proved a downward Löwenheim–Skolem theorem for modal-extended languages.

Abstract

Given a class $\mathcal C$ of models, a binary relation ${\mathcal R}$ between models, and a model-theoretic language $L$, we consider the modal logic and the modal algebra of the theory of $\mathcal C$ in $L$ where the modal operator is interpreted via $\mathcal R$. We discuss how modal theories of $\mathcal C$ and ${\mathcal R}$ depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside $L$. We calculate such theories for the submodel and the quotient relations. We prove a downward L\"owenheim--Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.