Characterization and definability in modal first-order fragments

TL;DR

This paper aims to identify general conditions under which model theoretic characterization and definability theorems hold across various modal first-order fragments, enabling unified proofs and extending results to new logics.

Contribution

It proposes sufficient conditions for theorems to hold in a wide range of modal logics, allowing for unified proofs and new results in unstudied logics.

Findings

Identified sufficient conditions for characterization and definability theorems.

Unified proof framework for multiple modal logics.

Potential to extend results to previously uninvestigated logics.

Abstract

Model theoretic results such as Characterization and Definability give important information about different logics. It is well known that the proofs of those results for several modal logics have, somehow, the same 'taste'. A general proof for most modal logics below first order is still too ambitious. In this thesis we plan to isolate sufficient conditions for the characterization and definability theorems to hold in a wide range of logics. Along with these conditions we will prove that, whichever logic that meets them, satisfies both theorems. Therefore, one could give an unifying proof for logics with already known results. Moreover, one will be able to prove characterization and definability results for logics that have not yet been investigated. In both cases, it is only needed to check that a logic meets the requirements to automatically derive the desired results.