  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.
