A General (Uniform) Relational Semantics for Sentential Logics

TL;DR

This paper introduces a unified relational semantics framework for various sentential logics, extending classical algebraic duality to more general algebraic structures and providing completeness and correspondence results.

Contribution

It generalizes the Jónsson-Tarski representation to posets, semilattices, and lattices, enabling a uniform semantics for classical and non-classical logics.

Findings

Provides a choice-free construction of canonical extensions.

Establishes completeness via generalized Sahlqvist algorithms.

Achieves uniform semantics for diverse sentential logics.

Abstract

We present a general relational semantics framework which, by varying the axiomatization and components of the relational structures, provides a uniform semantics for sentential logics, classical and non-classical alike. The approach we take rests on a generalization of the J\'{o}nsson-Tarski representation (and duality) for Boolean algebras with operators to the cases of posets, semilattices, or bounded lattices (with, or without distribution) with quasi-operators. Completeness proofs rely on a choice-free construction of canonical extensions for the algebras in the quasivarieties of the equivalent algebraic semantics of the logics. Correspondence results for axiomatic extensions of the logics of implication that we study rely on a fully abstract translation into their modal companions and they are calculated using a generalized Sahlqvist - van Benthem algorithm.