This unique monograph explores the cardinal, or quantitative, aspects of objects in the presence of vagueness, called vaguely defined objects. In the first part of the book such topics as fuzzy sets and derivative ideas, twofold fuzzy sets, and flow sets are concisely reviewed as typical mathematical representations of vaguely defined objects. Also, a unifying, approximative representation is presented. The second part uses this representation, together with Lukasiewicz logic as a basis for constructing a complete, general and easily applicable nonclassical cardinality theory for vaguely defined objects. Applications to computer and information science are discussed. Audience: This volume will be of interest to mathematicians, computer and information scientists, whose work involves mathematical aspects of vagueness, fuzzy sets and their methods, applied many-valued logics, expert systems and data bases.