 In any case, the vagueness of concepts has been much debated in philosophy (see the entry on  vagueness)  and it was one of the major motivations for the development of fuzzy logic (see the entry on  fuzzy logic).  In the 1960s, Lotfi Zadeh (1965) introduced the notion of a fuzzy set. A characteristic function of a set \(X\) is a mapping which is defined on a superset \(Y\) of \(X\) and which indicates membership of an element in \(X\). The range of the characteristic function of a classical set \(X\) is the two-element set \(\{0,1\}\) (which may be seen as the set of classical truth values). The function assigns the value 1 to elements of \(X\) and the value 0 to all elements of \(Y\) not in \(X\). A fuzzy set has a membership function ranging over the real interval [0,1]. A vague predicate such as ‘is much earlier than March 20th, 1963’, ‘is beautiful’, or ‘is a heap’ may then be regarded as denoting a fuzzy set. The membership function \(g\) of the fuzzy set denoted by ‘is much earlier than March 20th, 1963’ thus assigns values (seen as truth degrees) from the interval [0, 1] to moments in time, for example \(g\)(1p.m., August 1st, 2006) \(= 0\), \(g\)(3a.m., March 19th, 1963) \(= 0\), \(g\)(9:16a.m., April 9th, 1960) \(= 0.005\), \(g\)(2p.m., August 13th, 1943) \(= 0.05\), \(g\)(7:02a.m., December 2nd, 1278) \(= 1\).