Partial functions, Higher-order fuzzy logic, Fuzzy type theory, EQ-algebra

This paper is a study of fuzzy type theory (FTT) with partial functions. Out of several possibilities we decided to introduce a special value * which represents "undefined". In the interpretation of FTT, this value lays outside of the corresponding domain. In the syntax, it is naturally represented by the description operator acting on the empty (fuzzy) set which, of course, has no element and so, choosing an element from its kernel gives no result, i.e., it is undefined. We will demonstrate that our approach leads to reasonable characterization of the undefinedness. We will also show that any consistent theory of FTT has a model.