Proportoids
Christian Antić
TL;DR
Proportoids establish a rigorous framework for 4-ary analogical proportions $a:b::c:d$ on sets, with reflexivity, symmetry, and inner symmetry as core axioms and additional properties that tailor the structure. The paper develops proportional homomorphisms, congruences, and polymorphisms, connecting them through kernels and strong proportion-preserving principles, and extends the theory to partial analogies, proportional identities, and function relations. It also introduces various derived notions—diamond, join, triangular, bowtie, and square equivalences—capturing how mappings align under proportional reasoning, and applies these ideas to circles and similarity, all within a universal-algebra/predicate-logic perspective. The framework provides tools for analyzing analogical reasoning across algebraic and logical structures, with potential implications for formal models of analogy and inference.
Abstract
Analogical proportions are expressions of the form ``$a$ is to $b$ what $c$ is to $d$'' at the core of analogical reasoning. This paper contributes to the mathematical foundations of analogical proportions in the axiomatic tradition as initiated by Yves Lepage two decades ago. For this we introduce proportoids as sets endowed with a 4-ary analogical proportion relation $a:b::c:d$ satisfying a suitable set of axioms and study different kinds of proportion-preserving mappings and relations and their properties.