On fuzzy conditionals generalising the material conditional

Abstract

The paper deals with the problem of analysing if some Fuzzy Relations, widely used as Implications or Conditionals, are or are not good generalizations of the classical Material Conditionals. It is showed:

• 1) That Material Conditionals can be obtained from some elementary Fuzzy Preorders using special non-classical Membership Functions.

• 2) That other two large families of usual Fuzzy Conditionals both give the Material Conditionals only through Characteristic Functions, and that they are W-Fuzzy Preorders but neither Prod nor Min-Fuzzy Preorders.

• 3) That a third and very large family of Fuzzy Relations, sometimes considered in the literature as Implications, also generalizes the Material Conditionals when and only when are refined by Characteristic Functions.

• 4) That the family of Mamdani's (symmetrical and transitive) Fuzzy Conditionals generalize the restricted Material Conditionals when and only when Characteristic Functions are used.