The logic of quasi MV star algebras

TL;DR

The paper addresses the algebraic and logical underpinnings of quasi-MV* algebras by introducing quasi-Wajsberg* algebras as a term-equivalent variant and developing a dedicated logical system, q$\L^{*}$, grounded in valuations on $R^*=[-1,1]\times[-1,1]$. It establishes the core axioms (QW*1)-(QW*12) and demonstrates the crucial term-equivalence between quasi-Wajsberg* algebras and quasi-MV* algebras, including the mutual correspondences $f$ and $g$. The authors construct the semantic framework with $v^{*}$-valuations, prove soundness of the logic, and show that syntactic provability induces an MV*-algebra structure on the corresponding quotient, thereby linking the syntactic system to algebraic semantics. These results lay groundwork for completeness results and broaden the algebraic understanding of complex fuzzy logic and related quantum computation-inspired logics.

Abstract

Quasi-MV* algebras were introduced as generalizations of MV*-algebras and quasi-MV algebras. The recent investigation into quasi-MV* algebras shows that they are closely related to quantum computational logic and complex fuzzy logic. In this paper, we aim to study the logical system associated with quasi-MV* algebras in detail. First, we introduce quasi-Wajsberg* algebras as the term equivalence of quasi-MV* algebras and investigate the related properties of quasi-Wajsberg* algebras. Then we establish the logical system associated with quasi-Wajsberg* algebras using fewer deduction rules. Finally, we discuss the soundness of this logical system.

Theorems & Definitions (76)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Example 2.4
• Example 2.5
• Lemma 2.6
• Lemma 2.7
• Definition 3.1
• Example 3.2
• Remark 3.3