Concepts are the most fundamental unit of cognition that carries certain meanings in expression, thinking, reasoning, and system modeling. In denotational mathematics, a concept is formally modeled as an abstract and dynamic mathematical structure that encapsulates attributes, objects, and relations. The most important property of an abstract concept is its adaptive capability to autonomously interrelate itself to other concepts. This article presents a formal theory for abstract concepts and knowledge manipulation known as “concept algebra.” The mathematical models of concepts and knowledge are developed based on the object-attribute-relation (OAR) theory. The formal methodology for manipulating knowledge as a concept network is described. Case studies demonstrate that concept algebra provides a generic and formal knowledge manipulation means, which is capable to deal with complex knowledge and software structures as well as their algebraic operations.
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In cognitive informatics, logic, linguistics, psychology, software engineering, and knowledge engineering, concepts are identified as the basic unit of both knowledge and reasoning (Anderson, 1983; Colins & Loftus, 1975; Ganter & Wille, 1999; Hampton, 1997; Hurley, 1997; Matlin, 1998; Murphy, 1993; Wang, 2006a, 2006b, 2006c, 2007a, 2007c; Wang & Wang, 2006; Wilson & Keil, 1999). The rigorous modeling and formal treatment of concepts are at the center of theories for knowledge presentation and manipulation (Smith & Medin, 1981; Wille, 1982; Murphy, 1993; Codin, Missaoui, & Alaoui, 1995; Wilson & Keil, 1999; Yao, 2004; Chen & Yao, 2005). A concept in linguistics is a noun or noun-phrase that serves as the subject of a to-be statement (Hurley, 1997; Wang, 2002a, 2006a, 2006c, 2007d). Concepts in cognitive informatics (Wang, 2002a, 2006c, 2007b, 2007e) are an abstract structure that carries certain meaning in almost all cognitive processes such as thinking, learning, and reasoning.
Based on concepts and their relations, meanings of real-world concrete entities may be represented and semantics of abstract subjects may be embodied. Concepts can be classified into two categories, known as the concrete and abstract concepts. The former are proper concepts that identify and model real-world entities such as the sun, a pen, and a computer. The latter are virtual concepts that identify and model abstract subjects, which cannot be directly mapped to a real-world entity, such as the mind, a set, and an idea. The abstract concepts may be further classified into collective concepts, such as collective nouns and complex concepts, or attributive concepts such as qualitative and quantitative adjectives. The concrete concepts are used to embody meanings of subjects in reasoning while the abstract concepts are used as intermediate representatives or modifiers in reasoning.
A concept can be identified by its intension and extension (Hurley, 1997; Smith & Medin, 1981; Wang, 2006c; Wille, 1982; Yao, 2004).
For example, the intension of the concept pen connotes the attributes of being a writing tool, with a nib, and with ink. The extension of the pen denotes all kinds of pens that share the common attributes as specified in the intension of the concept, such as a ballpoint pen, a fountain pen, and a quill pen.