Abstract:

In the seminal paper of Zadeh about Fuzzy Logic \cite{zadeh}, 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 rational interval and finite chains.

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