Axiomatization of a Fundamental System for Generalized Ambiguous Four–Degree Membership Function Set Theory - Ali Aji; Bello Inuwa Mshelia; Yusuf Haruna (University of Mataram, 2026)
This paper presents the construction of an axiomatic system of Generalized Ambiguous Set Theory (AGAST)founded upon a four-valued logic, where the membership relation accepts four discrete degrees: True (α_A (x)), Partially True (β_A (x)), Partially False (γ_A (x)), and Falsity (η_A (x)) membership degree functions, The contribution of this work is the rigorous adaptation of foundational Zermelo-Frankel set theory (ZFC) and Axiomatic Fuzzy Set Theory principles, including the Axiom of Extensionality and the Axiom Schema of Ambiguous Separation, to cohere within the four-degree semantics. Furthermore, the theory introduces the Anti-Classicality Axiom, which postulates the existence of sets exhibiting non-classical membership degrees.