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:双关系框架上的无矛盾哥德尔模态逻辑。本文进一步发展了无矛盾哥德尔模态逻辑。在本文中,我们考虑带有Kripke语义的版本,该语义应用于具有两个模糊关系$R^+$和$R^-$(断言的真实程度和否定的真实程度)以及两个赋值$v_1$和$v_2$(真实性的支持和虚假性支持)的$[0,1]$ 值框架,并与De Morgan否定 $\neg$ 相关联。我们证明了它\emph{不}扩展哥德尔模态逻辑,且$\Box$和$\lozenge$不能相互定义。我们还展示了几个重要的框架类别是$\birelKGsquare$可定义的(特别是,脆性,单关系和有限分支)。对于$finitely$ $branching$框架上的$\birelKGsquare$,我们创建了一个声音和完整的约束表格演算和一个基于它的决策过程。使用决策程序,我们表明$\birelKGsquare$的可满足性和有效性都在PSPACE内。