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Concepts in Fuzzy Scaling Theory: order and granularity

机译：模糊缩放理论中的概念：顺序和粒度

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This paper explains mainly by examples the mathematical foundation of Fuzzy Scaling Theory described in (Wolff: EUFIT' 98, 6th European Congress on Intelligent Techniques and Soft Computing, Vol. I, Aachen, 1998, pp. 555-562). It illustrates the solution of fundamental problems in Fuzzy Theory, especially the problem of the pragmatic meaning of ordered sets as logics, the problem of the meaning of Fuzzy implications and the problem of the construction of direct products of linguistic variables. The key to the solution of these problems is the description of membership functions by cut contexts and of linguistic variables by conceptual seales. This renders the translation of Conceptual Scaling Theory into Fuzzy Scaling Theory and the generalization of the latter to L-Fuzzy Scaling Theory for an arbitrary ordered set L as logic. This paper bridges the gap between two scientific communities both using "graded concepts" as tools to handle granularity: it is shown that L-Fuzzy Scaling Theory is equivalent to Conceptual Scaling Theory.
机译：本文主要通过示例解释了模糊缩放理论的数学基础（Wolff：EUFIT'98，第六届欧洲智能技术和软计算大会，第一卷，亚琛，1998年，第555-562页）。它说明了模糊理论中基本问题的解决方案，特别是有序集作为逻辑的语用意义问题，模糊含意的含义问题和语言变量直接乘积的构造问题。解决这些问题的关键是通过剪切上下文来描述隶属函数，并通过概念上的密封来描述语言变量。这将概念缩放理论转换为模糊缩放理论，并将后者推广为L-模糊缩放理论，以任意有序集L为逻辑。本文弥合了两个科学界之间的鸿沟，它们都使用“分级概念”作为处理粒度的工具：表明L-模糊缩放理论等同于概念缩放理论。

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