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.

翻译：类比比例是形如“$a$ 之于 $b$ 正如 $c$ 之于 $d$”的表达式，构成类比推理的核心，而类比推理本身又是人类与人工智能的核心。作者近期从基本原理出发，在泛代数与一阶逻辑的通用框架内引入了一种抽象的代数逻辑类比比例体系。在该框架中，源代数与目标代数的底层语言完全相同。本文旨在将作者提出的单语框架拓展至双语框架，允许底层语言存在差异。这一目标通过在比例论证中引入修饰语来实现。该研究成果实现重大泛化，极大扩展了底层框架的适用性。从更广义的角度看，本文进一步推进了类比推理的数学理论构建。