A correspondence is established between one-variable fragments of (first-order) intermediate logics defined over a fixed countable linear frame and Gödel modal logics defined over many-valued equivalence relations with values in a closed subset of the real unit interval. It is also shown that each of these logics can be interpreted in the one-variable fragment of the corresponding constant domain intermediate logic, which is equivalent to a Gödel modal logic defined over (crisp) equivalence relations. Although the latter modal logics in general lack the finite model property with respect to their frame semantics, an alternative semantics is defined that has this property and used to establish co-NP-completeness results for the one-variable fragments of the corresponding intermediate logics both with and without constant domains.

在固定可数线性框架上定义的（一阶）中间逻辑的单变量片段与在具有实单位区间的封闭子集中的值的多值等价关系上定义的哥德尔模态逻辑之间建立了对应关系。还表明，这些逻辑中的每一个都可以在相应的常数域中间逻辑的单变量片段中解释，这相当于定义在（清晰）等价关系上的哥德尔模态逻辑。尽管后一种模态逻辑通常缺乏关于其框架语义的有限模型属性，但定义了具有此属性的替代语义，并用于为相应中间逻辑的单变量片段建立共同NP完整性结果有和没有常数域。