Analogical Proportions

Analogy-making is at the core of human and artificial intelligence and creativity. This paper introduces from first principles an abstract algebraic framework of analogical proportions of the form `$a$ is to $b$ what $c$ is to $d$' in the general setting of universal algebra. This enables us to compare mathematical objects possibly across different domains in a uniform way which is crucial for AI-systems. The main idea is to define solutions to analogical equations in terms of maximal sets of algebraic justifications, which amounts to deriving abstract terms of concrete elements from a `known' source domain which can then be instantiated in an `unknown' target domain to obtain analogous elements. It turns out that our notion of analogical proportions has appealing mathematical properties. We compare our framework with two recently introduced frameworks of analogical proportions from the literature in the concrete domains of sets and numbers, and we show that in each case we either disagree with the notion from the literature justified by some counter-example or we can show that our model yields strictly more solutions. As we construct our model from first principles using only elementary concepts of universal algebra, and since our model questions some basic properties of analogical proportions presupposed in the literature, to convince the reader of the plausibility of our model we show that it can be naturally embedded into first-order logic via model-theoretic types, and prove that analogical proportions are compatible with structure-preserving mappings from that perspective. This provides strong evidence for its applicability. In a broader sense, this paper is a first step towards a theory of analogical reasoning and learning systems with potential applications to fundamental AI-problems like commonsense reasoning and computational learning and creativity.