# Generalizing Analogical Inference from Boolean to Continuous Domains

arXiv: https://arxiv.org/abs/2511.10416 · HTML: https://mlsanity.com/paper/2511.10416/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 3 of its 28 references that are in the map.
- First posted on arXiv 13 Nov 2025.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 3 of 3)

- [Analogical Dissimilarity: Definition, Algorithms and Two Experiments in Machine Learning](https://mlsanity.com/paper/1401.3427/) (2014)
- [Towards Analogy-Based Explanations in Machine Learning](https://mlsanity.com/paper/2005.12800/) (2020)
- [FAME: Flexible, Scalable Analogy Mappings Engine](https://mlsanity.com/paper/2311.01860/) (2023)

## Built on by (most cited first, 2 of 2)

- [FrameNet Semantic Role Classification by Analogy](https://mlsanity.com/paper/2603.19825/) (2026)
- [From numerical proportions to analogical proportions between probabilities](https://mlsanity.com/paper/2606.23029/) (2026)

## Abstract (arXiv)

Analogical reasoning is a powerful inductive mechanism, widely used in human cognition and increasingly applied in artificial intelligence. Formal frameworks for analogical inference have been developed for Boolean domains, where inference is provably sound for affine functions and approximately correct for functions close to affine. These results have informed the design of analogy-based classifiers. However, they do not extend to regression tasks or continuous domains. In this paper, we revisit analogical inference from a foundational perspective. We first present a counterexample showing that existing generalization bounds fail even in the Boolean setting. We then introduce a unified framework for analogical reasoning in real-valued domains based on parameterized analogies defined via generalized means. This model subsumes both Boolean classification and regression, and supports analogical inference over continuous functions. We characterize the class of analogy-preserving functions in this setting and derive both worst-case and average-case error bounds under smoothness assumptions. Our results offer a general theory of analogical inference across discrete and continuous domains.