Dual-Domain Probability Valuation: A Proposed Mathematical Framework for Standard and Hypothetical Events

Sagar Verma Independent Researcher India

Preprint draft – July 11, 2026

Abstract

Classical probability assigns values in the interval [0, 1] to events in a specified sample space. In practical modelling, however, researchers also discuss event descriptions that lie outside the chosen sample space: omitted outcomes, counterfactual scenarios, physically unusual possibilities, and logically inconsistent statements. This paper proposes a formal separation between these classes. Starting from a standard probability space (Ω, F, P), we introduce a model-relative hypothetical description space H and a logically inconsistent class K. The dual domain is the disjoint union D = F ⊔ H. A dual-domain valuation L is defined by L(A) = P(A) for standard events A ∈ F, and L(h) = −d(h) for hypothetical descriptions h ∈ H, where d(h) ∈ (0, 1) is a bounded hypothetical-distance score derived from a transparent feature map. Logically inconsistent descriptions are excluded from the domain of L; an optional totalized representation assigns them the symbol ⊥, not the empty set. We establish range, reduction, monotonicity, boundary, stability, and metric results. We also prove a finite existence theorem and explain why the construction is generally not a signed probability measure. Worked coin and die examples illustrate how the method can be used as a model-audit and scenario-ranking tool. The manuscript deliberately removes an earlier zero-integral normalization proposal because no natural reference measure or scientific justification has yet been supplied. The resulting framework is therefore presented as a probability-inspired event valuation, open to calibration, empirical validation, and further axiomatic development.

Keywords: probability foundations; hypothetical events; negative valuation; model-relative uncertainty; event classification; distance function; signed measures.