We further develop the paraconsistent G\"{o}del modal logic. In this paper, we consider its version endowed with Kripke semantics on $[0,1]$-valued frames with two fuzzy relations $R^+$ and $R^-$ (degrees of trust in assertions and denials) and two valuations $v_1$ and $v_2$ (support of truth and support of falsity) linked with a De Morgan negation $\neg$. We demonstrate that it \emph{does not} extend G\"{o}del modal logic and that $\Box$ and $\lozenge$ are not interdefinable. We also show that several important classes of frames are $\birelKGsquare$ definable (in particular, crisp, mono-relational, and finitely branching). For $\birelKGsquare$ over finitely branching frames, we create a sound and complete constraint tableaux calculus and a decision procedure based upon it. Using the decision procedure we show that $\birelKGsquare$ satisfiability and validity are in PSPACE.

翻译：我们进一步发展了亚相容哥德尔模态逻辑。本文考虑其在[0,1]值框架上带有两个模糊关系R⁺和R⁻（对肯定和否定的信任度）以及通过德摩根否定¬相关联的两个赋值v₁和v₂（真值支持和假值支持）的克里普克语义版本。我们证明它并未扩展哥德尔模态逻辑，且□与◇不可相互定义。我们还展示了若干重要框架类（特别是清晰框架、单关系框架与有限分支框架）可由双关系Kripke哥德尔模态逻辑（记为birelKGsquare）定义。针对有限分支框架上的birelKGsquare，我们构建了可靠且完备的约束表列演算系统及其判定过程。利用该判定过程，我们证明了birelKGsquare的可满足性与有效性均属于PSPACE复杂度类。