Evaluations as L-Subsets

Abstract

In “Pure Mathematics” a statement like x ∈ X is either true or false, in formal terms: it has a truth value, “yes” or “no”, in numerical terms: tv(x ∈ X) ∈ L = {0, 1}. But in “Applied Mathematics” multivalued parameters are used. For example, if a given refrigerant is ecologically worthwhile may be answered by its values of ODP (ozone depletion potential), GWP (general warming potential) and ALT (atmospheric lifetime), i.e. by a triple of real numbers. Let us call such answers evaluations. They are elements of a lattice L, in this particular case of L = [0, 1]³, if the parameters are normalized. And we can consider such evaluations as L-subsets (see below) of the cartesian product O × A of the set O of objects and the set A of attributes. The crucial point is that this allows a choice of a suitable set theory together with the corresponding logic, in a problem-orientable way, since we can use a more or a less strict argumentation.

Dedicated to Professor Peter Paule on the occasion of his 60th birthday