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, disjunctive concepts of a relation, disjunctive 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.