Inspired by an interesting quotation from the literature, we propose four modalities, called ‘sane belief’, ‘insane belief’, ‘reliable belief’ and ‘unreliable belief’, and introduce logics with each operator as the modal primitive. We show that the four modalities constitute a square of opposition, which indicates some interesting relationships among them. We compare the relative expressivity of these logics and other related logics, including a logic of false beliefs from the literature. The four main logics are all less expressive than the standard modal logic over various model classes, and the logics of sane and insane beliefs are, respectively, equally expressive as the logics of unreliable and reliable beliefs on any class of models. The logics of reliable and unreliable beliefs are then combined into a bimodal logic, which turns out to be equally expressive as the standard modal logic. Despite this, we cannot obtain a complete axiomatization of the minimal bimodal logic, by simply translating the axioms and rules of the minimal modal logic |$\textbf{K}$| into the bimodal language. We then introduce a schematic modality which unifies reliable and unreliable beliefs and axiomatize it over the class of all frames and also the class of serial frames. This line of research is finally extended to unify sane and insane beliefs and some axiomatizations are given.

中文翻译：

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（疯狂）和（不可靠）信念的逻辑

受文献引述的有趣启发，我们提出了四种模式，分别称为“理智的信念”，“理智的信念”，“可靠的信念”和“不可靠的信念”，并介绍了以每个运算符为模态原语的逻辑。我们表明，这四个模式构成了一个对立的正方形，这表明它们之间存在一些有趣的关系。我们比较了这些逻辑和其他相关逻辑（包括错误信念的逻辑）的相对表达性从文学上。四种主要逻辑在各种模型类别上的表达均不如标准模态逻辑，并且理性信念和疯狂信念的逻辑分别具有与任何类别模型上不可靠和可靠信念的逻辑相同的表现力。然后将可靠和不可靠信念的逻辑组合成双峰逻辑，事实证明它与标准模态逻辑具有同等的表现力。尽管如此，仅通过翻译最小模态逻辑| $ \ textbf {K} $ |的公理和规则，我们就无法获得最小双峰逻辑的完整公理化。进入双峰语言。然后，我们介绍一种示意性模式，该模式将可靠和不可靠的信念统一起来，并在所有帧的类别以及串行帧的类别上公理化它。最终，这一研究领域扩展到统一理智和理智的信念，并给出了一些公理化方法。