A Measure-Theoretic Derivation of Zadeh Fuzzy Logic

Abstract

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.