When \((S,N)\)-implications are \((T,T _{1})\)-conditional functions?
One of the key issues in the formalization of approximate reasoning is the so-called compositional rule of inference, also called the generalized modus ponens, i.e., from an imprecise fact \(P\) and an imprecise rule \(P\Rightarrow Q\) can we deduce the imprecise conclusion \(Q\)? The answer heavily depends on the implication used to model the rule and on the triangular norm used to model the confluence between the given fact and the given rule. In mathematical terms, the following problem arises. Starting from a non-strict Archimedean t-norm \(T\), a continuous Archimedean t-conorm \(S\), and a strong negation induced by \(T\), under which conditions is the \([0,1]^2-[0,1]\) mapping \(T_1\), defined by \(T_1(x,y)=T(x,S(N(x),y))\), a triangular norm? This paper gives three characterizations.
scientific article; zbMATH DE number 1891695