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Top1. Introduction
A concept is a cognitive unit to identify and model a real-world concrete entity and a perceived-world abstract subject. The formal model and its discourse of abstract concepts are perceived widely different in literature (Tarski, 1944; Chomsky, 1956; Montague, 1974; Smith & Medin, 1981; Wille, 1982; Medin & Shoben, 1988; Partee et al., 1990; Murphy, 1993; Crystal, 1995; Codin et al., 1995; Bender, 1996; Pullman, 1997; Zadeh, 1999; Ganter & Wille, 1999; Wilson & Keil, 1999; O’Grady & Archibald, 2000; Taylor, 2002; Wang, 2007c, 2008b, 2009a, 2009b, 2010c, 2012c, 2013c, 2014a, 2014b; Wang et al., 2012; Tain et al., 2009, 2011; Hu et al., 2010) due to the lack of rigorous syntactic, semantic, and mathematical theories on formal concepts. Concepts in linguistics are a noun or noun-phrase that serves as the subject of a to-be statement (O’Grady & Archibald, 2000; Wang, 2007c, 2008b; Wang & Berwick, 2012, 2013). Concepts in cognitive informatics are an abstract structure that carries certain meaning in almost all cognitive processes such as thinking, learning, and reasoning (Wang, 2002, 2003, 2006, 2007b, 2007e, 2008a, 2009c, 2009d, 2009e, 2010a, 2011a, 2011b, 2012b, 2012d, 2012e, 2012f, 2012g, 2012h, 2013b; Wang & Wang, 2006; Wang & Chiew, 2010; Wang & Fariello, 2012; Wang et al., 2006, 2009a, 2009b, 2013). A typical perception on a general concept is a pair of intension and extension (Wille, 1982; O’Grady & Archibald, 2000; Wang, 2008b). Concepts in semantic algebra (Wang, 2013a) are formally modeled as a carrier of semantics (Keenan, 1975; Berners-Lee et al., 2001; Saeed, 2009; Wang, 2010b, 2010c; Wang et al., 2011) for a certain conceptual entity. In knowledge engineering and ontology, the rigorous modeling and treatment of concepts are at the center of theories for knowledge representation and manipulation (Smith & Medin, 1981; Wille, 1982; Murphy, 1993; Codin et al., 1995; Wang, 2009b, Wang & Tian, 2013; Bancroft & Wang. 2011).