An Ontological and Epistemological Perspective of Fuzzy Set Theory

Chapter 11

COMPENSATORY “AND”^*

In this chapter, we show that the break down of classical equivalences, i.e., FDCF ⊆ FCCF, together with the containment, i.e., FDCF ⊆ FCCF for strict and nilpotent Archimeidan t-norm and t-conorm provide a representation and an explanation for the “compensatory ‘AND’”.

It was stated earlier that linguistic operators “AND”, “OR”, etc., do not correspond one-to-one, isomorphically to t-norms and t-conorms. Experimental studies conducted by Zimmerman and Zysno (1980,1985) clearly validates the fact that linguistic “AND”, “OR” do not correspond to the t-norms and t-conorms. This is treated in this chapter with FDCF and FCCF structures.

It is shown that Zimmerman and Zysno’s γ operator takes on values within the FDCF ...