We develop a general theory of FDE-based modal logics. Our framework takes into account the four-valued nature of FDE by considering four partially defined modal operators corresponding to conditions for verifying and falsifying modal necessity and possibility operators. The theory comes with a uniform characterization for all obtained systems in terms of FDE-style formula-formula sequents. We also develop some correspondence theory and show how Hilbert-style axiom systems can be obtained in appropriate cases. Finally, we outline how different systems from the literature can be expressed in our framework.

中文翻译：

---

基于 $$ {FDE}$$FDE 的模态逻辑的通用框架

我们开发了基于 FDE 的模态逻辑的一般理论。我们的框架通过考虑对应于验证和伪造模态必要性和可能性算子的条件的四个部分定义的模态算子来考虑 FDE 的四值性质。该理论根据 FDE 式公式 - 公式序列对所有获得的系统进行了统一的表征。我们还开发了一些对应理论，并展示了如何在适当的情况下获得希尔伯特式公理系统。最后，我们概述了如何在我们的框架中表达与文献不同的系统。