We discuss a combinatorial interdependence between the maximal number of nonredundant symbols on n objects, if these elements of a set are looked at as a commutative and – at the same time – as a sequenced assembly. We generate combinations of partitions on a set (= "structures on a set"; = "multidimensional partitions") and compare the maximal number of distinct structures on a set (= "n?") to the maximal number of distinct sequences that the elements of the set can be ordered into (= "n!"). We find that max(n?,n!)=f(n). Switching between the longitudinal and the transversal way of storing information appears to allow inroads towards understanding some puzzling phenomena in a wide variety of fields. Specifically, we propose an explanation why FOUR distinct logical markers are used in the DNA by a number theoretical innovation. In the present paper, we concentrate on an important combinatorial detail which allows classifying (categorising, grouping) sets according to the maximal extent of inner disjunction of their subsets. We renumber the set N into a System M. We are still with additions of natural numbers (partitions), but select some, for which the translation function m(n) yields also .t. For these, m(n) = Sum( m(ni) ) above and beside n = Sum( ni ). We look into possible uses of .d. (doubly /or more/ .t.) sentences on N. Categories and Subject Descriptors: Combinatorics, number theory, disjunction operator, symmetry definitions, size-invariant translations. Key Words and Phrases: truth level .i. > 1; dissimilarity of parts of the whole, natural constants, extent of possibly being otherwise