Perfect paradefinite algebras are De Morgan algebras expanded with a perfection (or classicality) operation. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and single-conclusion (Set-Fmla) order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix, studied in depth by Gomes et al. (2022) from both the algebraic and the proof-theoretical perspectives. We continue hereby that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand such algebra with this connective and study the (self-extensional) Set-Set and Set-Fmla order-preserving logics, as well as the T-assertional logics of the variety induced by the new algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil's `symmetric modal logic'. For the Set-Set logic, in particular, the axiomatization we obtain is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property.

翻译：完美悖论代数是在德摩根代数基础上扩展了完美性（或经典性）运算所构成的代数结构。此类代数构成一个与对合斯通代数簇项等价的代数簇。与之相关的多结论（Set-Set）与单结论（Set-Fmla）保序逻辑是由六值矩阵决定的非代数化自外延形式不一致与不确定逻辑，Gomes等人（2022）已从代数和证明论角度对此进行了深入研究。本文延续该研究，探讨通过引入蕴含连接词（即满足演绎分离定理的蕴含算子）对这些逻辑进行保守扩展的路径。我们首先考虑由极其简单且易处理的非确定性语义给出的逻辑，其蕴含连接词（孤立使用时）具有经典逻辑特性——但这些逻辑无法保持自外延性。随后考虑由六值完美悖论代数上的相对伪补运算实现的蕴含连接词。我们的策略是通过该连接词扩展该代数结构，并研究（自外延的）Set-Set与Set-Fmla保序逻辑，以及由新代数簇诱导的T-断言逻辑。我们为这一新代数簇及其对应逻辑提供了公理化体系，并与对称海廷代数类及Moisil的"对称模态逻辑"进行了平行比较。特别地，针对Set-Set逻辑，我们获得的公理化具有分析性。最后，我们研究了这些逻辑的内插性质，并得出结论：新代数簇具有Maehara融合性质。