Real, rational and finite semantics for fuzzy logics

In the seminal paper of Zadeh about Fuzzy Logic, he defined fuzzy sets over the real unit interval but in real systems applications fuzzy sets over rational numbers in [0, 1] or over finite sets of values of a linguistic variable were also used. In this paper we try to analyze the differences between these different possible sets of truth-values in the setting of residuated multiple-valued logics underlying fuzzy sets. Moreover this analysis also clarifies the differences between t-norms in the strict sense (over [0, 1]) and t-norm like over other linearly ordered sets, namely the rationa linterval and finite chains.