Generalization-based similarity

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.