Detecting and exploiting similarities between seemingly distant objects 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 its plausibility, we show that it can be naturally embedded into first-order logic via model-theoretic types.

翻译：检测表面上差异巨大的对象之间的相似性并加以利用，是类比推理的核心，而类比推理本身又是人工智能的核心。本文从零开始，基于“泛化集合编码了元素的重要性质”这一观察，构建了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令人感兴趣的数学性质。由于我们仅使用泛代数的基本概念从第一性原理出发构建相似性概念，为向读者证明其合理性，我们展示了这种相似性可以通过模型论类型自然地嵌入到一阶逻辑中。