Algebraic modal correspondence: Sahlqvist and beyond

TL;DR

This paper offers a pedagogical and algebraic approach to Sahlqvist correspondence in modal logic, extending the theory beyond traditional classes and emphasizing its algebraic and order-theoretic foundations.

Contribution

It provides a uniform, modular, and elementary presentation of Sahlqvist correspondence, extending it to formulas outside the classical class and highlighting its algebraic and order-theoretic aspects.

Findings

Extended Sahlqvist correspondence to broader formula classes

Emphasized algebraic and order-theoretic foundations

Facilitated understanding of nonclassical logics' developments

Abstract

The present paper proposes a new introductory treatment of the very well known Sahlqvist correspondence theory for classical modal logic. The first motivation for the present treatment is {\em pedagogical}: classical Sahlqvist correspondence is presented in a uniform and modular way, and, unlike the existing textbook accounts, extends itself to a class of formulas laying outside the Sahlqvist class proper. The second motivation is {\em methodological}: the present treatment aims at highlighting the {\em algebraic} and {\em order-theoretic} nature of the correspondence mechanism. The exposition remains elementary and does not presuppose any previous knowledge or familiarity with the algebraic approach to logic. However, it provides the underlying motivation and basic intuitions for the recent developments in the Sahlqvist theory of nonclassical logics, which compose the so-called unified correspondence theory.