The Principle of Bounded Fuzziness for Uncertainty in Fuzzy Sets

Abstract

The representation of epistemic uncertainty is a fundamental issue in fuzzy set theory and approximate reasoning. Existing frameworks accommodate multiple admissible assessments but provide little insight into the structural form that epistemic uncertainty should assume. This paper introduces the Principle of Bounded Fuzziness (PBF), which postulates that admissible assessments generated under shared knowledge remain confined to a bounded neighborhood around a representative assessment in an appropriate metric space. Specializing the PBF to the space of membership functions [0,1]^X yields Ball Fuzzy Sets, defined as Chebyshev metric balls that admit a canonical three-way decomposition into belief, epistemic uncertainty, and disbelief satisfying the conservation relation b(x) + u(x) + d(x) = 1. We develop the structural and algebraic foundations of ball fuzzy sets, establish their principal geometric properties, derive complement, intersection, and union operations, and examine their relationships with interval-valued, hesitant, and higher-order fuzzy representations. Empirical studies on human probability distributions and expert medical assessments provide evidence that bounded disagreement is an observable phenomenon. In particular, finite coverage radii exist at multiple confidence levels; assessments derived from shared knowledge exhibit significantly tighter clustering than random baselines (p < 0.001); the estimated bounds remain stable under resampling; and large deviations become increasingly unlikely. These observations are consistent with the central premise that shared knowledge constrains admissible disagreement. An ICU risk assessment study illustrates the practical utility of the proposed framework for uncertainty-aware reasoning. By providing a geometric interpretation of epistemic uncertainty, the Principle of Bounded Fuzziness complements existing principles of uncertainty invariance and justifiable granularity.