We propose two classes of doxastic extensions of fuzzy \L ukasiewicz logic that are sound and complete with respect to some appropriate classes of Kripke-based models in which both atomic propositions and accessibility relations are fuzzy. One class of these extensions is equipped with pseudo-classical belief that has properties similar to the classical belief, and the other class is based on a new notion of belief that we call it \textit{skeptical} belief. We model a fuzzy version of the muddy children problem using pseudo-classical belief and a CPA-security experiment using skeptical belief, then by showing that the pseudo-classical belief is not appropriate for modeling the belief of an adversary in a CPA-experiment we justify proposing the notion of skeptical belief. Furthermore, we prove the soundness and completeness theorems for some of the proposed doxastic extensions.

翻译：我们提出了两类Doxastic扩展的模糊Łukasiewicz逻辑，这些逻辑在某些适当的基于Kripke的模型类中是完备而无误的，其中原子命题和可达性关系都是模糊的。这些扩展中的一类配备了类似于经典信念的伪古典信念，并且另一类是基于一个我们称之为“怀疑”信念的新概念。我们使用伪古典信念模拟了模糊版的“泥地孩子问题”，使用怀疑信念模拟了CPA安全实验，然后通过展示伪古典信念对于建模CPA实验中对手的信念不合适，我们证明了提出怀疑信念的必要性。此外，我们证明了一些提出的Doxastic扩展的完备性定理。