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

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Servin, Christian; Kreinovich, Vladik; Kosheleva, Olga
Computer Engineering
article description
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.