On the Logical and Algebraic Aspects of Reasoning with Formal Contexts

TL;DR

This paper develops and unifies modal logic systems to represent and reason about formal and rough concepts within formal contexts, linking algebraic structures like double Boolean algebras to logical frameworks.

Contribution

It introduces two-sorted modal logics KB and KF, unifies them into BM, and characterizes double Boolean algebras through logical and algebraic methods.

Findings

Unified logical framework for formal and rough concepts

Characterization of double Boolean algebras via Boolean algebras

Potential extensions for fine-grained reasoning in formal contexts

Abstract

A formal context consists of objects, properties, and the incidence relation between them. Various notions of concepts defined with respect to formal contexts and their associated algebraic structures have been studied extensively, including formal concepts in formal concept analysis (FCA), rough concepts arising from rough set theory (RST), and semiconcepts and protoconcepts for dealing with negation. While all these kinds of concepts are associated with lattices, semiconcepts and protoconcepts additionally yield an ordered algebraic structure, called double Boolean algebras. As the name suggests, a double Boolean algebra contains two underlying Boolean algebras. In this paper, we investigate logical and algebraic aspects of the representation and reasoning about different concepts with respect to formal contexts. We present two-sorted modal logic systems \textbf{KB} and \textbf{KF} for the representation and reasoning of rough concepts and formal concepts respectively. Then, in order to represent and reason about both formal and rough concepts in a single framework, these two logics are unified into a two-sorted Boolean modal logic \textbf{BM}, in which semiconcepts and protoconcepts are also expressible. Based on the logical representation of semiconcepts and protoconcepts, we prove the characterization of double Boolean algebras in terms of their underlying Boolean algebras. Finally, we also discuss the possibilities of extending our logical systems for the representation and reasoning of more fine-grained information in formal contexts.