Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gödel-Dummett and $\mathsf{BD_2}$: Linearity and Depth-Two Branching in Kripke Semantics

Vicent Navarro Arroyo

TL;DR

This work analyzes the semantic relationship between Gödel–Dummett logic GL and bounded-depth-2 logic BD_2 within intuitionistic Kripke semantics. It shows GL and BD_2 are incomparable by examining their minimal frame conditions: linear versus depth-bounded frames. The intersection of their frame classes exists, but their union GL+BD_2 collapses to frames of size at most two, making it nearly classical CPC. A conceptual interpretation ties GL to a single, forward-directed growth of knowledge while BD_2 embodies bounded deliberation, illustrating how orthogonal structural restrictions can force a classical collapse when combined.

Abstract

We study the semantic relationship between Gödel-Dummett logic $\mathsf{GL}$ and bounded-depth-2 logic $\mathsf{BD_2}$, two well-known intermediate logics. While $\mathsf{GL}$ imposes linearity on Kripke frames, $\mathsf{BD_2}$ bounds their depth to two. We prove these logics are incomparable (neither contains the other) through minimal frame conditions. Notably, their combination $\mathsf{GL+BD_2}$ collapses to the logic of one or two world frames, bringing it remarkably close to classical logic. This illustrates how controlling breadth and depth in intuitionistic semantics leads to mutually exclusive structural constraints. Finally, we give a conceptual and philosophical interpretation of the previous results. This is an extended abstract of work in progress. Comments and suggestions welcome at: vicent.navarro@ub.edu

Gödel-Dummett and $\mathsf{BD_2}$: Linearity and Depth-Two Branching in Kripke Semantics

TL;DR

This work analyzes the semantic relationship between Gödel–Dummett logic GL and bounded-depth-2 logic BD_2 within intuitionistic Kripke semantics. It shows GL and BD_2 are incomparable by examining their minimal frame conditions: linear versus depth-bounded frames. The intersection of their frame classes exists, but their union GL+BD_2 collapses to frames of size at most two, making it nearly classical CPC. A conceptual interpretation ties GL to a single, forward-directed growth of knowledge while BD_2 embodies bounded deliberation, illustrating how orthogonal structural restrictions can force a classical collapse when combined.

Abstract

We study the semantic relationship between Gödel-Dummett logic and bounded-depth-2 logic , two well-known intermediate logics. While imposes linearity on Kripke frames, bounds their depth to two. We prove these logics are incomparable (neither contains the other) through minimal frame conditions. Notably, their combination collapses to the logic of one or two world frames, bringing it remarkably close to classical logic. This illustrates how controlling breadth and depth in intuitionistic semantics leads to mutually exclusive structural constraints. Finally, we give a conceptual and philosophical interpretation of the previous results. This is an extended abstract of work in progress. Comments and suggestions welcome at: vicent.navarro@ub.edu
Paper Structure (8 sections, 6 theorems, 11 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Semantic Preliminaries
  3. Gödel--Dummett Logic
  4. The logic $\mathsf{BD_2}$
  5. Semantic separation and incomparability
  6. Conceptual interpretation
  7. Conclusion and future lines
  8. Appendix

Key Result

Theorem 3.1

Given a Kripke frame $\mathcal{F}=(W,\leq)$,

Figures (2)

  • Figure 1: Frame that satisfies $\mathsf{GL}$ but not $\mathsf{BD_2}$. Crossed out propositional letters indicate they are not forced at that world.
  • Figure 2: Frame that satisfies $\mathsf{BD_2}$ but not $\mathsf{GL}$. Crossed out propositional letters indicate they are not forced at that world.

Theorems & Definitions (19)

  • Definition 2.1: $\mathsf{IPC}$
  • Definition 2.2: $\mathsf{CPC}$
  • Definition 2.3: Kripke frames and models
  • Definition 2.4
  • Definition 3.1: $\mathsf{GL}$
  • Theorem 3.1: Frame condition for $\mathsf{GL}$
  • proof
  • Definition 4.1: $\mathsf{BD_2}$
  • Theorem 4.1: Frame condition for $\mathsf{BD_2}$
  • proof
  • ...and 9 more