Dear Lotfi:
I appreciate your idea about second-order multiplicity as a fruitful approach
to random sets. But there is an additional application, once you introduce the
proof of contradiction between a set in two different
appearances. You mention the idea that " the elements are sets of elements
drawn from a universe of discourse U". Exactly in this universe of discourse
some variables may appear negated in certain statements, like transforming your
example
L={({a, b}, 2), ({c, d}, 3)} being a second-order multiset, into
L={({-a, b}, 2), ({c, -d}, 3)}, which is also a second-order multiset.
Two decades ago I tried to follow this second approach in order to compare
together laws, as approved by a legislative body, with the aim to discover
contradictions among them. For a congressman working on a law proposal where
there is primarily a need to satisfy a certain need of a given social or
commercial or regional group. The large number of laws approved during one
legislation period is a hindrance in the task to compare it with former laws in
order to discover contradictions among them.
Should we succeed in formulating laws not only in plain text, but also in
logical formulas, the comparison would become more easy than before. Hence, I
suggest to you to explore for example the laws of the state of California
during the last legislative period and to find out if there is a reasonable
approach to formulate them as logical formulas. Indeed, there is a large
amount of experience from rule-oriented expert systems and Mamdani succeeded in
applying them to fuzzy sets.
I hope you can take some advantage from these ideas and find a new way to
apply multisets to a practical purpose.
Sincerely yours
Salomon Klaczko
D-12349 Berlin, Germany
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