From 1-D to 2-D Fuzzy: A Proof that Interval-Valued and Complex-Valued Are the Only Distributive Options

Authors

Christian Servin, El Paso Community CollegeFollow
Vladik Kreinovich, The University of Texas at El PasoFollow
Olga Kosheleva, The University of Texas at El PasoFollow

Publication Date

3-2015

Comments

Technical Report: UTEP-CS-15-21

Abstract

While the usual 1-D fuzzy logic has many successful applications, in some practical cases, it is desirable to come up with a more subtle way of representing expert uncertainty. A natural idea is to add additional information, i.e., to go from 1-D to 2-D (and multi-D) fuzzy logic. At present, there are two main approaches to 2-D fuzzy logic: interval-valued and complex-valued. At first glance, it may seem that many other options are potentially possible. We show, however, that, under certain reasonable conditions, interval-valued and complex-valued are the only two possible options.