Detecting and exploiting similarities is at the core of analogical reasoning which itself is at the core of artificial intelligence. This paper develops {\em from the ground up} an abstract algebraic and {\em qualitative} notion of similarity based on the observation that sets of generalizations encode important properties of elements. We show that similarity defined in this way has appealing mathematical properties. As we construct our notion of similarity from first principles using only elementary concepts of universal algebra, to convince the reader of the plausibility of our notion we show that it can be naturally embedded into first-order logic via model-theoretic types. In a broader sense, this paper is a further step towards a mathematical theory of analogical reasoning.

翻译：检测并利用相似性是类比推理的核心，而类比推理本身又是人工智能的核心。本文从基础出发，基于“泛化集合编码了元素的重要属性”这一观察，发展出一种抽象的代数且定性的相似性概念。我们证明了以这种方式定义的相似性具有吸引人的数学性质。由于我们仅使用泛代数的基本概念，从第一性原理构建了相似性概念，为了说服读者相信该概念的合理性，我们展示了它可以通过模型论类型自然地嵌入一阶逻辑。从更广泛的意义上说，本文是迈向类比推理数学理论的又一步。