扎德模糊逻辑的测度论推导

中文摘要 AI 辅助

扎德模糊逻辑通常通过赋予标量隶属度并规定标准补、最小合取和最大析取来引入。我们从一个严格正的、无原子的、归一化的布尔测度代数中推导出这一演算。测度的相等性将布尔补唯一地映射到$1-x$，但它不能将合取映射为单值运算：合取的精确标量像是重叠区间$[(x+y-1)^+,\min\{x,y\}]$。我们量化了标量化所损失的信息，证明了不存在非平凡的保测压缩能保留合取，并展示了重叠运算作为超运算是结合的。扎德合取$\min\{x,y\}$随后被强制为唯一非递减的幂等选择，并通过嵌套事件链全局实现；其德摩根对偶是扎德析取$\max\{x,y\}$。因此，一旦嵌套承诺被明确，最小-最大-标准否定演算就从测度论中得出。\Luk{}合取和乘积合取分别作为最小重叠和独立因子选择出现，澄清了替代的标量化。精确的极小极大结果、有序值扩展和三维实现障碍界定了标量模糊逻辑从布尔结构中保留的内容。

英文摘要

Zadeh fuzzy logic is normally introduced by assigning scalar membership degrees and postulating standard complement, minimum conjunction, and maximum disjunction. We derive this calculus from a strictly positive atomless normalized Boolean measure algebra. Equality of measure sends Boolean complement uniquely to $1-x$, but it cannot send meet to a single-valued operation: the exact scalar image of conjunction is the overlap interval $[(x+y-1)^+,\min\{x,y\}]$. We quantify the information lost under scalarization, prove that no nontrivial measure-preserving compression can retain meet, and show that the overlap operation is associative as a hyperoperation. Zadeh conjunction $\min\{x,y\}$ is then forced as the unique nondecreasing idempotent selection and is realized globally by nested event chains; its De Morgan dual is Zadeh disjunction $\max\{x,y\}$. Thus the min--max--standard-negation calculus follows from measure theory once the nesting commitment is made explicit. The \Luk{} and product conjunctions arise as minimum-overlap and independent-factor selections, clarifying alternative scalarizations. Exact minimax results, an ordered-valued extension, and a three-dimensional realization obstruction delimit what scalar fuzzy logic retains from Boolean structure.

发表机构

• Indraprastha Institute of Information Technology Delhi（德里印度理工学院）

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