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A General (Uniform) Relational Semantics for Sentential Logics

Chrysafis Hartonas

TL;DR

This work presents a unified, uniform relational semantics framework for a wide range of sentential logics by extending Jónsson–Tarski representations to posets, semilattices, and bounded lattices with quasi-operators, and by formulating choice-free canonical extensions. It develops sorted residuated frames and their duals to model implicative algebras, then derives complete, sound semantics via canonical models and modal translations, using a generalized Sahlqvist–van Benthem algorithm for correspondences. The Lambek calculus (non-associative and associative) and their substructural variants are given explicit frame axioms and canonical proofs, with detailed accounts for unit, weakening, contraction, and exchange. The framework is then extended to distributive, intuitionistic, and Boolean logics, including a fully general procedure (Steps 1–6) to obtain first-order correspondents for distribution-free modalities, offering a robust, choice-free methodology for relational semantics across a spectrum of logics. The resulting approach provides a principled bridge between algebraic and relational semantics, enabling modular completeness results and facilitating machine-checkable translations to modal and first-order formalisms.

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 Jónsson-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.

A General (Uniform) Relational Semantics for Sentential Logics

TL;DR

This work presents a unified, uniform relational semantics framework for a wide range of sentential logics by extending Jónsson–Tarski representations to posets, semilattices, and bounded lattices with quasi-operators, and by formulating choice-free canonical extensions. It develops sorted residuated frames and their duals to model implicative algebras, then derives complete, sound semantics via canonical models and modal translations, using a generalized Sahlqvist–van Benthem algorithm for correspondences. The Lambek calculus (non-associative and associative) and their substructural variants are given explicit frame axioms and canonical proofs, with detailed accounts for unit, weakening, contraction, and exchange. The framework is then extended to distributive, intuitionistic, and Boolean logics, including a fully general procedure (Steps 1–6) to obtain first-order correspondents for distribution-free modalities, offering a robust, choice-free methodology for relational semantics across a spectrum of logics. The resulting approach provides a principled bridge between algebraic and relational semantics, enabling modular completeness results and facilitating machine-checkable translations to modal and first-order formalisms.

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 Jónsson-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.

Paper Structure

This paper contains 34 sections, 33 theorems, 46 equations, 18 tables.

Table of Contents

  1. Introduction
  2. Implicative Algebras, Frames, Logics, and Canonical Extensions
  3. Logics of Ordered Algebras
  4. Sorted Residuated Frames with Relations
  5. The Underlying Polarity of a Frame -- The Lattices of Stable and Co-stable Sets
  6. The Dual Sorted Powerset and Full Complex Algebra of a Frame
  7. From Frames to Models
  8. Canonical Extensions
  9. Pure Implicative Logics
  10. Algebras, Languages and Logics
  11. Implicative Frames and Relational Semantics
  12. Representation, Canonical Extension and Completeness
  13. The Unit as a Left Identity Element
  14. Modal Translation and Correspondence
  15. Soundness and Completeness
  16. ...and 19 more sections

Key Result

Proposition 1

Let $\mathbf{P}$ be a poset, a semilattice or a lattice and let $\mathfrak{F}=\mathbf{P}_+$ be its associated frame $(s,W,I)$ where $W_1=\hbox{\rm Filt}(\mathbf{P}), W_\partial=\hbox{\rm Idl}(\mathbf{P})$ and $xIy$ iff $x\cap y=\emptyset$. Let ${\mathcal{G}}(W_1)$ be the complete lattice of Galois s

Theorems & Definitions (78)

  • Definition 1
  • Definition 2
  • Remark 1: Kripke, Routley-Meyer and Goldblatt Frames
  • Remark 2
  • Definition 3
  • Remark 3
  • Remark 4
  • Definition 4
  • Remark 5
  • Definition 5
  • ...and 68 more