Intensional Inheritance Between Concepts: An Information-Theoretic Interpretation

2025-01-29

AI Summary

This paper addresses the quantitative modeling of “intensional inheritance” between concepts—specifically, the information conveyed by the presence of property F about the likelihood of property W holding for an object. We formalize intensional inheritance uniformly as an information transmission problem, establishing the first general computational framework grounded in Shannon information theory and probabilistic semantics. We prove that extensional inheritance is a special case of this formulation. Leveraging mutual information to quantify property dependence, we derive a general formula for intensional inheritance and obtain a closed-form solution under the mutual exclusivity assumption for properties. Furthermore, we establish an explicit analytical relationship between the conditional probability (P(W mid F)) and mutual information. This work introduces a novel paradigm for the quantitative analysis of conceptual semantic relations, enabling principled, information-theoretic characterization of intensional dependencies among properties.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning
• Machine Learning: Information Theory

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

This paper addresses the problem of formalizing and quantifying the concept of"intensional inheritance"between two concepts. We begin by conceiving the intensional inheritance of $W$ from $F$ as the amount of information the proposition"x is $F$"provides about the proposition"x is $W$. To flesh this out, we consider concepts $F$ and $W$ defined by sets of properties $left{F_{1}, F_{2}, ldots, F_{n} ight}$ and $left{W_{1}, W_{2}, ldots, W_{m} ight}$ with associated degrees $left{d_{1}, d_{2}, ldots, d_{n} ight}$ and $left{e_{1}, e_{2}, ldots, e_{m} ight}$, respectively, where the properties may overlap. We then derive formulas for the intensional inheritance using both Shannon information theory and algorithmic information theory, incorporating interaction information among properties. We examine a special case where all properties are mutually exclusive and calculate the intensional inheritance in this case in both frameworks. We also derive expressions for $P(W mid F)$ based on the mutual information formula. Finally we consider the relationship between intensional inheritance and conventional set-theoretic"extensional"inheritance, concluding that in our information-theoretic framework, extensional inheritance emerges as a special case of intensional inheritance.

Problem

• Information Transfer
• Conceptual Inheritance
• Semantic Relations

Innovation

• Information Theory
• Inherence Inheritance
• Extensional Inheritance