Author: Matthew T. Armendariz
December 12, 2025
INTRODUCTORY NOTE
This paper argues that what the AI field calls “hallucination” in large language models is better understood through Kripke’s rule-following paradox. LLMs learn from finite training data and extrapolate to novel inputs. This is structurally identical to the problem Kripke identified in Wittgenstein on Rules and Private Language: any finite set of examples is compatible with multiple rules, and nothing in the data alone determines which rule the learner has internalized. When the learned function diverges from truth on a novel input, the model has no internal error signal. It is, in Kripke’s terms, computing “quus” rather than “plus.”
The paper defines this phenomenon as “ungrounded divergence” and identifies three architectural features of LLMs that guarantee its occurrence: (1) finite and fixed training corpora, (2) no access to ground truth during generation, and (3) stochastic output selection. Together, these features make divergence ineliminable under current architectures, regardless of scale or fine-tuning.
The framework has direct implications for legal practice and other high-stakes domains. The paper evaluates Retrieval-Augmented Generation (RAG) as a partial mitigation strategy, framing it as a form of Kripkean rigid designation that re-grounds the model’s outputs in authoritative sources. It also critiques recent technical proposals (including logprob monitoring and multi-model consensus systems) that claim to detect or eliminate hallucination, arguing that these approaches fail to address the structural problem the framework identifies.