Intensional Inheritance Between Concepts: An Information-Theoretic Interpretation

TL;DR

The paper addresses formalizing intensional inheritance between concepts $F$ and $W$ through information-theoretic measures. It develops both Shannon and algorithmic information theory formulations, incorporating interaction information among properties to quantify $I(F;W)$ and the conditional $P(W|F)$. A tractable special case with mutually exclusive properties yields simple expressions and demonstrates how extensional inheritance arises as a special case when properties are singleton. The framework provides a quantitative tool for reasoning about concept hierarchies in AI systems, linking intensional and extensional inheritance under a unified information-theoretic view.

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 $\left\{F_{1}, F_{2}, \ldots, F_{n}\right\}$ and $\left\{W_{1}, W_{2}, \ldots, W_{m}\right\}$ with associated degrees $\left\{d_{1}, d_{2}, \ldots, d_{n}\right\}$ and $\left\{e_{1}, e_{2}, \ldots, e_{m}\right\}$, 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.