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Dick Ballard said:
The intuition you are looking for is in the physical notion of "degrees of freedom". One seeks to enumerate all the possible situations or geometries or conditional interactions that might lead to a physically distinct reality. You can picture your n-dimensional space as being defined by n independent variables, and for each variable a set of instances over which each ranges. A single n-ary relationship connects together one instance each from all n variables. So each n-ary represents one possible reality, and the bundle of "all" relationships, describes every reality thought physically meaningful.
Comment from Paul Werbos
I have assumed that the "Bayesian" kind of reasoning described by Michael Jordan, for example, is rooted in exactly such n-ary relations, as the "cliques" or "elites" of a kind of Markov Random Field of binary propositions. Markov Random Fields, by the way, are equally important in the true foundations of quantum field theory.