We introduce a two-sort weighted modal logic for possibilistic reasoning with fuzzy formal contexts. The syntax of the logic includes two types of weighted modal operators corresponding to classical necessity ($\Box$) and sufficiency ($\boxminus$) modalities and its formulas are interpreted in fuzzy formal contexts based on possibility theory. We present its axiomatization that is \emph{sound} with respect to the class of all fuzzy context models. In addition, both the necessity and sufficiency fragments of the logic are also individually complete with respect to the class of all fuzzy context models. We highlight the expressive power of the logic with some illustrative examples. As a formal context is the basic construct of formal concept analysis (FCA), we generalize three main notions in FCA, i.e., formal concepts, object oriented concepts, and property oriented concepts, to their corresponding $c$-cut concepts in fuzzy formal contexts. Then, we show that our logical language can represent all three of these generalized notions. Finally, we demonstrate the possibility of extending our logic to reasoning with multi-relational fuzzy contexts, in which the Boolean combinations of different fuzzy relations are allowed.

翻译：我们提出了一种用于模糊形式背景可能性推理的双类加权模态逻辑。该逻辑的语法包含两类加权模态算子，分别对应于经典必然性（$\Box$）模态和充分性（$\boxminus$）模态，其公式基于可能性理论在模糊形式背景中进行解释。我们给出了该逻辑的公理化系统，该系统相对于所有模糊背景模型类是**可靠**的。此外，该逻辑的必然性片段和充分性片段分别相对于所有模糊背景模型类也是完备的。我们通过一些示例说明了该逻辑的表达能力。由于形式背景是形式概念分析（FCA）的基本构造，我们将FCA中的三个主要概念——形式概念、对象导向概念和属性导向概念——推广到它们在模糊形式背景中对应的$c$-截概念。然后，我们证明了我们的逻辑语言能够表示所有这三种推广后的概念。最后，我们展示了将该逻辑扩展至多关系模糊背景推理的可能性，其中允许不同模糊关系的布尔组合。