AN INTERPRETATION OF MEMBERSHIP FUNCTIONS AND THE PROPERTIES OF GENERAL PROBABILISTIC OPERATORS AS FUZZY SET OPERATORS .2. EXTENSION TO 3-VALUED AND INTERVAL-VALUED FUZZY-SETS

Previously in Part I of the present paper and its supplement an approa ch was made to an interpretation of membership functions as probabilit y and a proposal of the probabilistic operators with dependence relati on of the sets. This paper is an extension to three-valued and interva l-valued fuzzy sets. By taking into consideration uncertainty and cont radiction in the judgment whether or not an element of universe of dis course belongs to a fuzzy subset, upper, lower, and intermediate grade s of membership are defined; this definition leads to a notion of thre e-valued fuzzy set, and when only uncertainty is considered interval-v alued fuzzy set results. Under the proposed set operation rules, the p robabilistic operators with dependency, various basic set operations a nd their properties are studied. Of the properties satisfied by the cr isp set operators many of them are found to be valid. Some properties, e.g., excluded middle laws are found to be invalid due to the uncerta inty and contradiction involved. Also discussed are a representation o f set operations of type 2 fuzzy sets by the three-valued fuzzy set sy stem, and the connection of the proposed operators with other typical set operators. (C) 1997 Elsevier Science B.V.