Problems with the defuzzification method and a new representation using Lukasiewicz logic

We show that the common defuzzification method has some major deficiencies in regard to certain formal logical properties among rules like AND-equivalence, contrapositive-equivalence, chaining-property, etc. To correct this problem, we present a new method for representing a function y=g(x) using a rule set based on Lukasiewicz´s logic L_ℵ. The method introduces two sets of normalized fuzzy concepts {A_j: 1⩽j⩽N} and {C_j: 1⩽j⩽N}, a set of rules of the form r_j=“if x is A_j then y is C_j” or “if x is ~A_j then y is C_j” where A_j and C_j have non-decreasing linear membership functions, and an AND/OR formula φ(r₁, r₂, ..., r_N) over the rules r_i, without involving negation, which describes how to combine the results of inference from the individual rules r_j to obtain the final value of y. Our rules satisfy the logical properties in regard to the various equivalences when applicable