New Framework for Determinization in Plural Structure Theories

The paper introduces a formal framework for creating canonical interpretations from plural structure theories, which are defined by a signature, axioms, and an inference policy. It categorizes non-determinism into epistemic plurality (Type E) and structural plurality (Type S), with a further refined Type S-strong subclass. Two primary mechanisms for canonicalization are explored: operator-based completion and selector-based construction. The research provides conditions under which these mechanisms exist, showing how pure inference-based completion can reduce to a saturated closure operator under specific rules. The framework also examines multi-level canonicalization as a non-commutative system and offers a classification theorem. Notably, it applies these concepts to Large Language Model (LLM)-assisted reasoning, suggesting that phenomena like hallucination can be understood as unsupported canonicalization within this theoretical context.