A Modal Logic for Possibilistic Reasoning with Fuzzy Formal Contexts
Prosenjit Howlader, Churn-Jung Liau
TL;DR
The paper introduces a two-sort weighted modal logic (2WML) for possibilistic reasoning over fuzzy formal contexts, linking FCA concepts with cut-based modalities under possibility theory. It defines formal semantics, soundness for the full 2WML, and completeness for its necessity (2WKB) and sufficiency (2WKF) fragments, including finite-degree restrictions. It further shows how $c$-cut formal, object-oriented, and property-oriented concepts can be represented within 2WML and demonstrates extension to multi-relational fuzzy contexts (2WBML) with Boolean combinations of relations. The work lays foundational semantics and axiomatizations for reasoning about uncertain, fuzzy FCA data, with potential applications in knowledge discovery and data mining under uncertainty, and outlines avenues for future extensions to multi-valued, probabilistic, and toolbox-implementations.
Abstract
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.