Unbiased Canonical Set-Valued Oracles Via Lattice Theory

Computer Science > Artificial Intelligence [Submitted on 24 Jun 2026] Unbiased Canonical Set-Valued Oracles Via Lattice Theory Jobst Heitzig A non-agentic "oracle" AI that estimates probabilities of future events faces a self-reference problem: once its answer is learned and acted upon, it can change the very probability it was asked to report. One response, advocated for the Scientist AI programme, is to ask only counterfactual questions, evaluated as if the answer had no influence. We observe that such answers tend to become irrelevant the moment they are learned, precisely because their premise is then false. We therefore explore a self-referential alternative in which the oracle reports not a single probability but a credal set that is simultaneously unbiased and self-consistent with the consequences of being learned. The naive self-consistency requirement is satisfied by too many sets (including the useless answer [0,1]), so the problem is to single out a canonical, nontrivial member. We do so with the Knaster--Tarski fixed-point theorem on the complete lattice of closed credal sets, taking the least fixed point of a suitably defined isotone operator; a variant instead reports the least fixed point that contains every self-consistent point estimate. We prove existence, self-consistency, and nonemptiness, show that the construction collapses to the classical point answer for non-performative questions, and that for a binary event the canonical answer is, under a natural hull-factoring assumption, an interval. The development is purely lattice-theoretic and extends unchanged from a binary event B to an arbitrary random variable X, with P(B\mid A,C) replaced by the conditional law \mathcal{L}(X\mid A,C). We close with open questions, including whether the interval characterization itself survives that generalization. Comments: 18 pages Subjects: Artificial Intelligence (cs.AI); Machine Learning (cs.LG) MSC classes: 68T01 ACM classes: I.2 Cite as: arXiv:2606.26418 [cs.AI]   (or arXiv:2606.26418v1 [cs.AI] for this version)   https://doi.org/10.48550/arXiv.2606.26418 Focus to learn more Submission history From: Jobst Heitzig [view email] [v1] Wed, 24 Jun 2026 22:20:13 UTC (19 KB) Access Paper: HTML (experimental) view license Current browse context: cs.AI < prev   |   next > new | recent | 2026-06 Change to browse by: cs cs.LG References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)