GSS–URUT: Rule Underdetermination and the Undecidability of Semantic Determination

This paper develops the framework of Generative Structural Semantics (GSS) and presents a formal version of the Rule Underdetermination Theorem (URUT). Within GSS, semantic determination is not taken to arise directly from structural facts, but through a process of rule selection. A given structure may be compatible with multiple candidate rules, implying that structural information alone is insufficient to uniquely determine semantic outcomes. To formalize this problem, the paper introduces the Rule Sufficiency Problem (RSP), defined as the problem of deciding whether a rule yields a total binary-valued decision procedure for all inputs. By reduction from the Halting Problem and by diagonal arguments based on Kleene’s Recursion Theorem, RSP is shown to be undecidable. The paper further argues that, under an Effective Canonicalization assumption, semantic determination requires the effective selection of a canonical rule from a space of admissible rules. Since this process depends on solving instances of RSP, and RSP is undecidable, semantic determination is shown to be non-computable. The result is not presented as a reduction of Gödel’s incompleteness theorem, Turing’s halting undecidability, or Tarski’s undefinability of truth. Rather, URUT is proposed as exhibiting a structurally analogous limitation arising from self-reference, diagonalization, and totality constraints. The framework aims to provide a computational perspective on the relation between structure and meaning, and offers a formal approach to questions concerning vagueness, rule-following, and semantic foundations.