Strong quasi-MV* algebras and their logics
Lei Cai, Wenjuan Chen
TL;DR
This work develops an algebraic framework for complex fuzzy logic by introducing the subvariety ${\mathbb{SQMV^*}}$ of strong quasi-MV$^\ast$ algebras and showing its term-equivalence with strong quasi-Wajsberg$^\ast$ algebras. It proves a representation theorem: every strong quasi-MV$^\ast$ algebra embeds into a direct product of an MV$^\ast$-algebra and a flat strong quasi-MV$^\ast$ algebra, with standard completeness achieved via the canonical flat model ${\textbf{F}}({\textbf{MV^*}_{[-1,1]}},0)$; canonical square and disk models ${\textbf{S^*}}$, ${\textbf{D^*}}$ share the same equational theory. The paper then introduces a corresponding logic ${\text{sq}\L^{\ast}}$ (sqL$^{\ast}$) and proves soundness and completeness with respect to the strong quasi-Wajsberg$^\ast$ semantics, clarifying ties to the classical $\L^*$ and Wajsberg$^\ast$ frameworks. By establishing functorial equivalences and concrete models, it provides a solid algebraic and proof-theoretic foundation for CFL-related logics and their vagueness-aware semantics.
Abstract
In this paper, we introduce the subvariety of quasi-MV* algebras in order to characterize the logic which is related to complex fuzzy logic. First, we give the definitions of strong quasi-MV* algebra and strong quasi-Wajsberg* algebra and show that they are term equivalence. Second, we present the representation theorem and the standard completeness of strong quasi-MV* algebras. Moreover, we discuss the properties of terms in the language of Wajsberg* algebras and strong quasi-Wajsberg* algebras. Finally, we establish the logical system associated with strong quasi-Wajsberg* algebra and prove that the logical system is sound and complete.