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We discuss the idea of fuzzy relationships and their role in modeling weighted social relational networks. The paradigm of computing with words is introduced, and the role that fuzzy sets play in representing linguistic concepts is described. We discuss how these technologies can provide a bridge between a network analyst's linguistic description of social network concepts and the formal model of the network. We then turn to some examples of taking an analyst's network concepts and formally representing them in terms of network properties. We first do this for the concept of clique and then for the idea of node importance. Finally, we introduce the idea of vector-valued nodes and begin developing a technology of social network database theory.