From Fuzzy Values to Intuitionistic Fuzzy Values to Intuitionistic Fuzzy Intervals Etc.: Can We Get an Arbitrary Ordering?

Traditional fuzzy logic uses real numbers as truth values. This description is not always adequate, so in intuitionistic fuzzy logic, we use pairs of real numbers to describe a truth value. Such pairs can be described either as pairs (t; f) for which t + f 1, or, alternatively, as pairs (t; 1 ? f) for which t 1 ?f. To make this description even more adequate, instead of using real numbers to described each value t and f , we can use intervals, and thus get interval-valued intuitionistic fuzzy values which can be described by 4 real numbers each. We can iterate this procedure again and again. The question is: can we get an arbitrary partially ordered set in this manner? An arbitrary lattice? In this paper, we show that although we cannot thus generate arbitrary lattices, we can actually generate an arbitrary partially ordered set in this manner. In this sense, the \inter-valization" operation which underlies the notion of an intuitionistic fuzzy set, is indeed universal.