Bilingual analogical proportions via hedges
Christian Antić
TL;DR
The paper addresses extending analogical proportions from a unilingual to a bilingual algebraic framework by introducing hedges and a unified language to connect two languages L_A and L_B via a common index set. It defines L-justifications and maximal d-maximal arrow proportions across a pair of algebras A and B, and proves key invariance results, including the Bilingual Uniqueness Lemma and the Bilingual Functional Proportion Theorem. Notable contributions include the formalization of Jus_L(V), d-maximality, and characteristic L-justifications, along with preservation of the proportional axiom set. This work broadens the reach of mathematical analogical reasoning to cross-language settings with potential implications for AI, logic, and cognitive modeling.
Abstract
Analogical proportions are expressions of the form ``$a$ is to $b$ what $c$ is to $d$'' at the core of analogical reasoning which itself is at the core of human and artificial intelligence. The author has recently introduced {\em from first principles} an abstract algebro-logical framework of analogical proportions within the general setting of universal algebra and first-order logic. In that framework, the source and target algebras have the {\em same} underlying language. The purpose of this paper is to generalize his unilingual framework to a bilingual one where the underlying languages may differ. This is achieved by using hedges in justifications of proportions. The outcome is a major generalization vastly extending the applicability of the underlying framework. In a broader sense, this paper is a further step towards a mathematical theory of analogical reasoning.