Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Formalising Propositional Information via Implication Hypergraphs

Vibhu Dalal

TL;DR

The paper presents a framework to quantify the information content of logical propositions using implication hypergraphs, arguing that informativeness arises from the proposition's implications to other propositions. It defines the propositional information vector $I_{\nu,\epsilon}$ as the fixed-point solution of $I = A I + \epsilon A 1 + \nu l$, with $I = (I - A)^{-1} (\epsilon A 1 + \nu l)$ when $1$ is not an eigenvalue of $A$, and leaves $\nu$ and $\epsilon$ as parameters to encode base information and incompleteness. The approach is illustrated on mathematical examples, discusses positivity, parameter roles, and limitations, and emphasizes that mathematics is an especially suitable testbed for the method. The authors sketch directions for refinement, including improved handling of conjunctions, nonuniform contribution among premises, and broader applicability beyond pure mathematics.

Abstract

This work introduces a framework for quantifying the information content of logical propositions through the use of implication hypergraphs. We posit that a proposition's informativeness is primarily determined by its relationships with other propositions; specifically, the extent to which it implies or derives other propositions. To formalize this notion, we develop a framework based on implication hypergraphs, that seeks to capture these relationships. Within this framework, we define propositional information, derive some key properties, and illustrate the concept through examples. While the approach is broadly applicable, mathematical propositions emerge as an ideal domain for its application due to their inherently rich and interconnected structure. We provide several examples to illustrate this and subsequently discuss the limitations of the framework, along with suggestions for potential refinements.

Formalising Propositional Information via Implication Hypergraphs

TL;DR

The paper presents a framework to quantify the information content of logical propositions using implication hypergraphs, arguing that informativeness arises from the proposition's implications to other propositions. It defines the propositional information vector as the fixed-point solution of , with when is not an eigenvalue of , and leaves and as parameters to encode base information and incompleteness. The approach is illustrated on mathematical examples, discusses positivity, parameter roles, and limitations, and emphasizes that mathematics is an especially suitable testbed for the method. The authors sketch directions for refinement, including improved handling of conjunctions, nonuniform contribution among premises, and broader applicability beyond pure mathematics.

Abstract

This work introduces a framework for quantifying the information content of logical propositions through the use of implication hypergraphs. We posit that a proposition's informativeness is primarily determined by its relationships with other propositions; specifically, the extent to which it implies or derives other propositions. To formalize this notion, we develop a framework based on implication hypergraphs, that seeks to capture these relationships. Within this framework, we define propositional information, derive some key properties, and illustrate the concept through examples. While the approach is broadly applicable, mathematical propositions emerge as an ideal domain for its application due to their inherently rich and interconnected structure. We provide several examples to illustrate this and subsequently discuss the limitations of the framework, along with suggestions for potential refinements.

Paper Structure

This paper contains 8 sections, 3 theorems, 21 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminary Definitions
  3. Propositional Information
  4. Applications
  5. The role of $\epsilon$ and $\nu$
  6. Examples
  7. Discussion
  8. Conclusion

Key Result

Proposition 3.1

Let $H\in \mathcal{H}$, and $\nu, \epsilon>0$. Let $A$ denote the adjacency matrix of $H=(V,E)$, where $V=\{p_1,\ldots,p_n\}$. Let $\mathbf{1}$ be the vector of $n$ ones, and $l$ be a vector where the $i$-th entry is 1 if $p_i$ is a leaf node, and 0 otherwise. If $A - I$ is invertible, or equivalent

Figures (6)

  • Figure 1: An example of an implication hypergraph with hyperedges $(\{p_1, p_2\}, \{p_3\})$, $(\{p_3\}, \{p_4\})$, and $(\{p_4\}, \{p_2\})$.
  • Figure 2: An example of a non-minimal implication hypergraph.
  • Figure :
  • Figure :
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (13)

  • Definition 2.1: Hypergraph
  • Definition 2.2: Implication Hypergraph
  • Definition 2.4: Minimal Implication Hypergraph
  • Definition 3.1: Adjacency Matrix
  • Definition 3.2: Propositional Information
  • Proposition 3.1
  • Definition 3.3: Configured Implication Hypergraph
  • Theorem 3.1
  • proof
  • Remark 3.1
  • ...and 3 more