A Frequency-Theoretic Construction of Possibility Measures: Axiomatic Foundations, Vagueness Quantification, and Engineering Decision Systems

Jayesh Vijay Rao Karanjgaonkar

Abstract: A frequency-based possibility distribution is constructed from empirical counts under a Unique Mode Condition (UMC). Given a discrete frequency distribution satisfying UMC, the natural possibility measure (where denotes the modal frequency) is shown to be the unique normalised mapping satisfying the possibility axioms, frequency proportionality, and singleton domination of the corresponding empirical probability ?that is, preserves frequency proportionality and dominates the corresponding singleton probabilities, a weaker and more carefully scoped claim than full event-level probability?possibility consistency. Vagueness measures based on the resulting probability and possibility representations are then developed: the Proximity Value (total surplus possibility), the Elemental Proximity Value (per-observation contribution), and the Mode Proximity Value (baseline vagueness), together with a frequency-based specificity measure and exact bidirectional conversion identities between and under UMC. PV is further shown to equal the -divergence . Illustrative engineering and decision datasets are used to examine the empirical behaviour of the proposed measures: the Proximity Value achieves Pearson correlations with human vagueness judgements, comparing favourably with fuzzy -means, rough sets, Bayesian intervals, and intuitionistic fuzzy entropy baselines (, Wilcoxon), and a decision procedure demonstrates their potential use in vagueness-aware control. The results indicate promising agreement with human vagueness assessments, while multimodality, small samples, discretisation, and cultural scope of the rating studies are identified as important limitations.

Keywords: possibility measure; vagueness quantification; proximity value; specificity; frequency distribution; hybrid human-machine systems; uncertainty quantification

How to Cite?: Jayesh Vijay Rao Karanjgaonkar, "A Frequency-Theoretic Construction of Possibility Measures: Axiomatic Foundations, Vagueness Quantification, and Engineering Decision Systems", Volume 15 Issue 9, September 2026, International Journal of Science and Research (IJSR), Pages: 683-689, https://www.ijsr.net/getabstract.php?paperid=SR26909085205, DOI: https://dx.doi.org/10.21275/SR26909085205

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