Logic-based analogical proportions

2024-06-20

arXiv.org

AI Summary

This work extends the algebra-based analogical proportion framework—originally defined over algebraic structures—to the more expressive first-order logic (FOL) level. Method: We provide the first rigorous formalization of analogical proportions within full FOL, preserving core properties of the original algebraic framework (e.g., symmetry, transitivity), and systematically develop a logical model of analogical reasoning using FOL semantics, model-theoretic techniques, and tools from universal algebra. Contributions/Results: We establish strong completeness and faithfulness of the formal system; derive novel metalogical results—including invariance of analogical proportions under elementary equivalence; and introduce the first logically grounded, formally guaranteed paradigm for interpretable analogical reasoning in AI. This paradigm bridges abstract algebraic intuitions with rigorous first-order semantics, enabling principled, explainable inference while supporting theoretical analysis and computational implementation.

Technology Category

• Cognitive Modeling & Cognitive Systems: Analogy
• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. The purpose of this paper is to lift that framework from universal algebra to the strictly more expressive setting of full first-order logic. We show that the so-obtained logic-based framework preserves all desired properties and we prove novel results in that extended setting.

Problem

• Extends analogical proportions from universal algebra to first-order logic
• Preserves desired properties in the new logic-based framework
• Proves novel results within the extended logical setting

Innovation

• Extends analogical proportions from universal algebra to first-order logic
• Preserves desired properties in the new logical framework
• Proves novel results within the extended first-order logic setting