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Hybrid logic for strict betweenness

Rafał Gruszczyński, Zhiguang Zhao

Abstract

The paper is devoted to modal properties of the ternary strict betweenness relation as used in the development of various systems of geometry. We show that such a relation is non-definable in a basic similarity type with a binary operator of possibility, and we put forward two systems of hybrid logic, one of them complete with respect to the class of dense linear betweenness frames without endpoints, and the other with respect to its subclass composed of Dedekind complete frames.

Hybrid logic for strict betweenness

Abstract

The paper is devoted to modal properties of the ternary strict betweenness relation as used in the development of various systems of geometry. We show that such a relation is non-definable in a basic similarity type with a binary operator of possibility, and we put forward two systems of hybrid logic, one of them complete with respect to the class of dense linear betweenness frames without endpoints, and the other with respect to its subclass composed of Dedekind complete frames.

Paper Structure

This paper contains 15 sections, 33 theorems, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Betweenness relations and linear orders
  3. Hybrid logic with binary operators
  4. Non-definability
  5. The tools
  6. The results
  7. Definability in extended languages
  8. The axiom of density with respect to LBWE
  9. The hybrid logic of strict betweenness
  10. The system Bh
  11. Completeness of Bh
  12. The system Bh+
  13. Completeness of Bhp
  14. Summary and future work
  15. Elementary properties of betweenness

Key Result

Theorem 4.1

A first-order definable class of binary Kripke frames is $\mathcal{H}(\mathop{\mathrm{@}}\nolimits)$-definable iff it is closed under taking ultrafilter morphic images and generated subframes.

Figures (3)

  • Figure 1: A bounded morphism $f$ which shows that outer transitivity for $B$ (expressed by \ref{['B4']}) is not modally definable. Betweenness relations hold between triples of points that are connected by the lines of the same style. The domain on the left is divided into four clusters. Each cluster is mapped onto precisely one point.
  • Figure 2: Failure of the modal density axiom in a non-dense frame. Since there is no point between $x$ and $y$, $x\nVdash\mathop{\mathrm{C}}\nolimits\mathop{\mathrm{C}}\nolimits p$
  • Figure 3: Failure of the modal density axioms entails failure of the geometrical density. Here is pictured the situation in which all points on the $y$ side of $x$ fail to satisfy $\mathop{\mathrm{C}}\nolimits p$.

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 3.1
  • Theorem 4.1: tenCate-MTFEML
  • Definition 4.2
  • Definition 4.3
  • Definition 4.4
  • Definition 4.5
  • Definition 4.6
  • Definition 4.7
  • Theorem 4.8: Goldblatt-Thomason-ACIPML
  • ...and 64 more