Similarity-based analogical proportions

TL;DR

This paper reframes analogical proportions through the lens of algebraic similarity within universal algebra, defining a proportion a:b::c:d via groundings in arrow-based similarity (Arr$(\mathfrak A)$) and cross-algebra generalizations. The approach embeds similarity at the core of proportions, allowing transfer of results across the two notions and enabling direct application of future similarity findings to analogical proportions. It provides a rigorous construction of the similarity-based proportion relation, analyzes which classical axioms hold or fail, and proves that isomorphisms preserve proportion relations, with a thorough comparison to prior frameworks. The work advances a unified, algebraically grounded view of analogy, offering a robust foundation for future theoretical and applied work in logic, program synthesis, and related areas.

Abstract

The author has recently introduced abstract algebraic frameworks of analogical proportions and similarity within the general setting of universal algebra. The purpose of this paper is to build a bridge from similarity to analogical proportions by formulating the latter in terms of the former. The benefit of this similarity-based approach is that the connection between proportions and similarity is built into the framework and therefore evident which is appealing since proportions and similarity are both at the center of analogy; moreover, future results on similarity can directly be applied to analogical proportions.

Theorems & Definitions (23)

• Definition 1
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• Definition 5
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• Definition 10
• Example 11
• Theorem 12