Formalising Propositional Information via Implication Hypergraphs

TL;DR

This paper proposes a novel framework using implication hypergraphs to quantify the information content of logical propositions based on their relationships and implications among each other.

Contribution

It introduces a formal framework for propositional information using implication hypergraphs, capturing the interconnectedness and implications among propositions.

Findings

Framework effectively models propositional relationships

Mathematical propositions are ideal for this approach

Framework has limitations and potential for refinement

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.