Modal reduction principles across relational semantics

TL;DR

This paper develops a unified theoretical framework to compare and relate the first-order correspondents of Sahlqvist modal reduction principles across various relational semantics, including crisp and many-valued frames.

Contribution

It introduces a systematic approach to interrelate and transfer properties of modal reduction principles across diverse relational semantic settings using unified correspondence theory.

Findings

Established connections among first-order correspondents in different relational contexts.

Identified conditions for encoding the same modal content across various structures.

Facilitated transfer of properties like reflexivity and transitivity across semantic frameworks.

Abstract

The present paper establishes systematic connections among the first-order correspondents of Sahlqvist modal reduction principles in various relational semantic settings which include crisp and many-valued Kripke frames, and crisp and many-valued polarity-based frames (aka enriched formal contexts). Building on unified correspondence theory, we aim at introducing a theoretical environment which makes it possible to: (a) compare and inter-relate the various frame correspondents (in different relational settings) of any given Sahlqvist modal reduction principle; (b) recognize when first-order sentences in the frame-correspondence languages of different types of relational structures encode the same "modal content"; (c) meaningfully transfer and represent well known relational properties such as reflexivity, transitivity, symmetry, seriality, confluence, density, across different semantic contexts. These results can be understood as a first step in a research program aimed at making correspondence theory not just (methodologically) unified, but also (effectively) parametric.