No Universal Hyperbola: A Formal Disproof of the Epistemic Trade-Off Between Certainty and Scope in Symbolic and Generative AI

TL;DR

This paper formally disproves a conjecture claiming a universal trade-off between certainty and scope in AI, using information theory and providing counterexamples and logical inconsistencies.

Contribution

It provides a rigorous mathematical disproof of the proposed epistemic trade-off, showing no universal hyperbola bounds certainty and scope in AI models.

Findings

The conjecture leads to internal inconsistency with prefix Kolmogorov complexity.

Counterexample refutes the conjecture with plain Kolmogorov complexity.

Entropy-based revision cannot restore the universal trade-off.

Abstract

In direct response to requests for a logico-mathematical test of the conjecture, we formally disprove a recently conjectured artificial intelligence trade-off between epistemic certainty and scope in its published universal hyperbolic product form, as introduced in Philosophy and Technology. Certainty is defined as the worst-case correctness probability over the input space, and scope as the sum of the Kolmogorov complexities of the input and output sets. Using standard facts from coding theory and algorithmic information theory, we show, first, that when the conjecture is instantiated with prefix (self-delimiting, prefix-free) Kolmogorov complexity, it leads to an internal inconsistency, and second, that when it is instantiated with plain Kolmogorov complexity, it is refuted by a constructive counterexample. These results establish a main theorem: contrary to the conjecture's claim, no universal "certainty-scope" hyperbola holds as a general bound under the published definitions. We further show that a subsequent "entropy-based" revision, replacing the Kolmogorov scope with Shannon joint entropy and redefining the epistemic certainty level accordingly, cannot restore universality either.