Analogical proportions
Christian Antić
TL;DR
This work develops an abstract algebraic framework for analogical proportions within universal algebra, enabling cross-domain reasoning by modeling $a:b::c:d$ via sets of algebraic justifications and term rewrites. It proves key results, including a Functional Proportion Theorem and isomorphism-compatible First and Second Isomorphism Theorems, and provides both a rigorous logical interpretation through model-theoretic rewrite types and practical instantiations in sets and arithmetic. The analysis shows that analogical proportions are local and non-monotonic, and it critically examines existing axioms, offering a refined axiom set. The paper also situates the framework relative to SMT, boolean and word proportions, and discusses potential AI applications and future directions, including algorithmic computation and extensions to richer logics. Overall, it establishes a principled, cross-domain theory of analogical reasoning with a clear path to AI-oriented applications in common-sense reasoning and creative problem solving.
Abstract
Analogy-making is at the core of human and artificial intelligence and creativity with applications to such diverse tasks as proving mathematical theorems and building mathematical theories, common sense reasoning, learning, language acquisition, and story telling. 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. It turns out that our notion of analogical proportions has appealing mathematical properties. 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 from that perspective that analogical proportions are compatible with structure-preserving mappings. This provides conceptual 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 common sense reasoning and computational learning and creativity.