标题： 通过蕴含超图形式化命题信息 显示英文标题

标题： Formalising Propositional Information via Implication Hypergraphs

摘要： 这项工作引入了一个框架，通过使用蕴含超图来量化逻辑命题的信息内容。我们提出，一个命题的有用性主要由它与其他命题之间的关系决定——特别是它能够蕴含或推导出其他命题的程度。为了形式化这一概念，我们开发了一个基于蕴含超图的框架，旨在捕捉这些关系。在该框架中，我们定义了命题信息，推导了一些关键性质，并通过示例说明了这一概念。虽然该方法具有广泛的应用性，但由于数学命题本身具有丰富且相互关联的结构，因此它们成为该方法应用的理想领域。我们提供了几个示例来说明这一点，随后讨论了该框架的局限性以及可能的改进建议。 显示英文摘要

摘要： 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.