# Unified Correspondence

## Abstract

The present chapter is aimed at giving a conceptual exposition of the mathematical principles underlying Sahlqvist correspondence theory. These principles are argued to be inherently algebraic and order-theoretic. They translate naturally on relational structures thanks to Stone-type duality theory. The availability of this analysis in the setting of the algebras dual to relational models leads naturally to the definition of an *expanded* (object) language in which the well-known ‘minimal valuation’ meta-arguments can be encoded, and of a *calculus for correspondence* of a proof-theoretic style in the expanded language, mechanically computing the first-order correspondent of given propositional formulas. The main advantage brought about by this formal machinery is that correspondence theory can be ported in a uniform way to families of nonclassical logics, ranging from substructural logics to mu-calculi, and also to different semantics for the same logic, paving the way to a uniform correspondence theory.

## Keywords

Sahlqvist correspondence theory Duality Algorithmic correspondence Intuitionistic modal logic mu-calculus## References

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