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Canonicity in power and modal logics of finite achronal width

Robert Goldblatt, Ian Hodkinson

TL;DR

A method for showing that various modal logics that are valid in their countably generated canonical Kripke frames must also be valid in their uncountably generated ones is developed.

Abstract

We develop a method for showing that various modal logics that are valid in their countably generated canonical Kripke frames must also be valid in their uncountably generated ones. This is applied to many systems, including the logics of finite width, and a broader class of multimodal logics of `finite achronal width' that are introduced here.

Canonicity in power and modal logics of finite achronal width

TL;DR

A method for showing that various modal logics that are valid in their countably generated canonical Kripke frames must also be valid in their uncountably generated ones is developed.

Abstract

We develop a method for showing that various modal logics that are valid in their countably generated canonical Kripke frames must also be valid in their uncountably generated ones. This is applied to many systems, including the logics of finite width, and a broader class of multimodal logics of `finite achronal width' that are introduced here.
Paper Structure (47 sections, 15 theorems, 48 equations, 6 figures)

This paper contains 47 sections, 15 theorems, 48 equations, 6 figures.

Table of Contents

  1. Keywords.
  2. Introduction
  3. Layout of paper
  4. Basic definitions
  5. General machinery used in the paper
  6. Multimodal logic
  7. Structures
  8. Kripke frames, models, semantics
  9. Point-generated inner subframes
  10. $\mathsf{M}$-BAOs and translations
  11. Canonical extensions
  12. Canonicity in power
  13. Strategy for canonicity
  14. The variety $V$
  15. The two-sorted structure $\mathfrak{B}$
  16. ...and 32 more sections

Key Result

PROPOSITION 3.3

Figures (6)

  • Figure 1: the ${\cal L}(\mathfrak{A})$-elementary maps $\delta,\iota,\sigma$. The diagram commutes.
  • Figure 2: The intransitive frame ${\cal D}_j$ validating $5_2$
  • Figure 3: the logics $KU_n$, $K4I_n$, $K5_2$, and $K5$ (monomodal case)
  • Figure 4: irreflexive transitive frame ${\cal G}_j$ for §\ref{['sss:KUn-K4In gap']} and §\ref{['sss:KU1-K52 gap']}
  • Figure 5: transitive frame ${\cal E}_j^n$ for §\ref{['sss:KUn+1-KUn']}
  • ...and 1 more figures

Theorems & Definitions (45)

  • DEFINITION 2.1
  • DEFINITION 2.2
  • DEFINITION 2.3
  • DEFINITION 2.4
  • DEFINITION 2.5
  • DEFINITION 3.1
  • DEFINITION 3.2
  • PROPOSITION 3.3
  • proof
  • COROLLARY 3.4
  • ...and 35 more