The Logical Operations on Relations, Scales and Concept Lattices

Object-attribute-value-relationships, which are a frequently used data structure to code real-world problems, are formalized via many-valued contexts in Formal Concept Analysis (FCA). The aim of FCA is to explore (formal) concepts from empirical data contexts. A concept of a context consists of two parts: the extent (objects the concept covers) and the intent (attributes describing the concept). From the logical point of view, the intent of each concept is a conjunction of some attributes. Similar to conjunction, negation and disjunction are also important logical operations of attributes or attribute-value pairs, which are common in human language. However, the classical FCA as a mathematical theory of concepts lacks a theory of negation and disjunction. In this paper, we take negation and disjunction into consideration in the process of constructing concepts, and hence obtain the following extended concepts: negative concepts of a context (i.e., a binary relation), negative concepts of a relation with some scales, V-concepts of a relation, V-concepts of a relation with some scales, logical concepts of a relation, and logical concepts of a relation with some scales. Compared with concepts in the classical FCA, the extended concepts mentioned above are more pertinent and more meaningful.