The IF THEN ELSE statement and interval-valued fuzzy sets of higher type

Abstract

A new fuzzy relation which represents an IF THEN ELSE (abbreviated to “ITE”) statement is constructed. It is shown that a relation which (a) always gives correct inference and (b) does not contain false information which is not present in the ITE statement, must be of a higher degree of fuzziness than the antecedents and consequents of the ITE statement. Three different ways of increasing the fuzziness of the relation are used here: (1) the fuzzy relation is of higher type; (2) it is interval-valued; (3) contains a BLANK or “don't know” component. These three types of fuzziness come about naturally because the relation is a restriction of an initial relation which represents the second approximation to the state of complete ignorance. There exist successive approximations to the state of complete ignorance, each of them being an intervalvalued fuzzy set BLANK of one type higher than the previous one. Similar representations of the zeroth and first approximation to the state of ignorance have been used in the theory of probability, though in a rather heuristic fashion. The assignment of a value to a variable is represented as a complete restriction of the BLANK state of type N; the “value” being any pure (non-interval-valued) fuzzy set of type N. With the new relation, the inferred set is a superposition of the consequents of the ITE statement and of the BLANK state, each of these components being multiplied by an interval-valued coefficient. In the case of modus ponens inference, the component with the highest coefficient (determined from a specially defined ordering relation for interval-values) is always the correct consequent, provided that the original ITE statement is logically consistent. A mathematical test for logical consistency of the (possibly fuzzy) ITE statement is given. Disjointness of fuzzy sets is defined and connected up with logical consistency. The paradoxes of the implication of mathematical logic disappear in the fuzzy set treatment of the ITE statement. Subnormal fuzzy singletons find their natural interpretation. When used as an antecedent in an ITE statement, such a singleton does not have enough strength to induce the consequent with complete certainty. Instead it induces a superposition of an interval-valued fuzzy set and the BLANK state. New definitions for union, intersection and complementation of fuzzy sets of higher type are suggested. A new interpretation of an interval-valued fuzzy set of type N as a collection of fuzzy sets of type N, not as a type N+1 set, is given. There exist two types of union, intersection and complementation for intervalvalued fuzzy sets. They are called the fuzzy and the crisp operations, respectively. It is suggested that the negation be represented by an interval-valued fuzzy set. We conclude that increased fuzziness in a description means increased ability to handle inexact information in a logically correct manner.