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Ultrafilter Extensions for Veltman Semantics

Felix Frigola Gonzalez, Joost J. Joosten, Vicent Navarro Arroyo, Cosimo Perini Brogi

Abstract

In this paper, we present a first-order frame condition for interpretability logic and show that the condition is not modally definable. Yet, the frame-condition holds both on ILM and on ILP frames and, hence, is of potential importance for the long-standing open problem about the interpretability logic of all reasonable arithmetical theories. In the light of the Goldblatt-Thomason Theorem, the modally inexpressible frame condition serves as motivation to develop ultrafilter extensions for inter pretability logic. We develop the necessary algebraic tools to define these ultrafilter extensions and prove the main properties about both the tools and the ultrafilter extensions.

Ultrafilter Extensions for Veltman Semantics

Abstract

In this paper, we present a first-order frame condition for interpretability logic and show that the condition is not modally definable. Yet, the frame-condition holds both on ILM and on ILP frames and, hence, is of potential importance for the long-standing open problem about the interpretability logic of all reasonable arithmetical theories. In the light of the Goldblatt-Thomason Theorem, the modally inexpressible frame condition serves as motivation to develop ultrafilter extensions for inter pretability logic. We develop the necessary algebraic tools to define these ultrafilter extensions and prove the main properties about both the tools and the ultrafilter extensions.
Paper Structure (15 sections, 32 theorems, 5 equations, 1 figure)

This paper contains 15 sections, 32 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Axiomatic Calculus and Relational Semantics for $\textup{IL}$
  4. Labelling Techniques for Interpretability Logics
  5. Ultrafilter Extensions
  6. A Non-Definability Result
  7. Ultrafilter Extensions
  8. Theories, Filters and Ultrafilters
  9. Ultrafilter Extensions for Frames and Models
  10. Full Labels in Algebras
  11. Elementary Equivalence and Modal Saturation
  12. Conclusions
  13. Detailed Proof of Lemma \ref{['notriralgebra']}
  14. Detailed proof of Lemma \ref{['labelextension']}
  15. Detailed proof of Lemma \ref{['extension_iff_translation']}

Key Result

Lemma 2.5

The assuring relation $\prec_S$ satisfies the following properties:

Figures (1)

  • Figure 1: $\mathcal{F}_0$ and $\mathcal{F}_1$ frames.

Theorems & Definitions (72)

  • Definition 2.1: Axiomatization of $\textup{IL}$
  • Definition 2.2: Veltman frames and models
  • Definition 2.3: Frame definability
  • Definition 2.4: $S$-assuring successor
  • Lemma 2.5: From GorisBilkovaJoostenMikec:2020:ArXivLabels, Lemma 4.2
  • Lemma 2.6: From GorisBilkovaJoostenMikec:2020:ArXivLabels, Lemma 4.6
  • Lemma 2.7: From GorisBilkovaJoostenMikec:2020:ArXivLabels, Lemma 3.7
  • Definition 2.8
  • Lemma 2.9: Ultrafilter Principle
  • Definition 2.10
  • ...and 62 more