Degrees that are not probabilities

Abstract

Organizations reason in vague predicates. A customer is large, an exposure is material, a supplier is strategic, and none of these has a threshold anyone can defend, which is why the thresholds that end up in software are invented in meetings and then never revisited. This paper describes the fuzzy inference capability of our analytics extension: linguistic variables, membership functions, expert-authored rules, and three inference families — Mamdani with defuzzification, Sugeno, and Tsukamoto — running as Suggestors inside the governed loop of our substrate. Two things make it a paper rather than a library note. The first is a distinction the implementation enforces in the type system: a membership degree and a materiality degree are both numbers in [0,1] and are not the same quantity, and neither is a probability. Confusing them is the same category error this series has now met three times, and the remedy is the same — make the compiler refuse. The second is a constraint that is genuinely mathematical rather than stylistic: Tsukamoto inference requires monotone consequent membership functions, because the firing strength must be inverted to a crisp value, and our implementation returns an error rather than an approximation for the triangular, trapezoidal and Gaussian shapes where the inverse does not exist.