Abstract
Compensatory Fuzzy Logic is a transdisciplinary axiomatic theory, different from the Classical Norm and Conorm approach to improving interpretability by natural language. Archimedean Compensatory Fuzzy Logic (ACFL), introduced recently, uses different properties and interpretations of involved truth values. Membership functions involved are not studied explicitly in fuzzy theories, even though it is essential in solving problems. The definition of parameterized families of membership functions is not rare in fuzzy literature. However, according to our review, each of those families has the same shape except the recently introduced Continuous Linguistic Variables. That has been a limitation in the expressiveness of linguistic values. Besides, except for Dombi's theory, these functions are often not related to logical operators. This paper aims to use ACFL to overcome each of these drawbacks. We generalize some fuzzy concepts, only using the ACFL generator function. A Generalized Sigmoidal Function and a Generalized Linguistic Modifier are s-shaped functions generated by it. Those elements define a parameterized family containing different shape functions like an increasing sigmoidal, decreasing sigmoidal and convex function; we call it a Generalized Continuous Linguistic Variable. This paper improves ACFL by unifying it into single theory elements like logic generator functions, linguistic modifiers, membership functions, and linguistic variables. The improved ACFL is not just a Pluralist Logic that makes compatible the classical approach of Norm and Conorm with CFL theory, but a contextual pluralist logic able to select a logic that better expresses specific contextual knowledge. This theory is valuable in Knowledge Discovery; because it creates new searching elements that allow selecting the `best logic ' for a particular dataset. We develop knowledge discovery cases for different databases to illustrate it and show its data sensitivity.