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Annex 3 of Representation, Comprehension and Communication of Sets: the Role of Number (1978) Originally published in International Classification, 6, 1979,2, pp. 92-103
If a set is named (e.g. "development"), the question may be asked in how many ways possible elements may be distinguished by subdividing the set.
2-level distinction: The set may, for example, be split into 2 subsets, but in how many ways may this be done in a particular case? Depending on the level at which the distinction is made, there may be 1, 2, 3, 4, or N recognized 2-level distinctions; namely the most fundamental, and successively less fundamental levels of distinction. Clearly these are not unrelated, since the less fundamental distinctions are regrouped in distinctions at more fundamental levels. For example, at the level at which only 4 distinctions can be recognised, the regrouping would tend to bear a relationship to the level at which only 8 distinctions are made (by regrouping pairs of distinctions). On initial examination of all such 2-level distinctions, there would tend to be some confusion as to the level to which they should be allocated in order that the most fundamental should not be embedded in a set of less fundamental distinctions. The probability of any particular 2-level distinction being advocated as most fundamental is likely to be higher, the greater the number of possible distinctions at that level. (Namely it is less likely that the more fundamental 2-level distinctions would be recognized.)
On the other hand this tendency is counter-balanced by the lower stability, viability and acceptability of the less fundamental distinctions. Over longer periods of time they are meaningful to fewer and are of less value to the ordering of perceptions, however vigorously the use of any particular one may be advocated.
In sorting out to which level each 2-level distinction belongs, reference may be made to the pattern of relations between the various distinctions at that level in the light of the underlying qualitative characteristics of the number associated with that level (see Annex 2, for example).
3-level distinction: The set may however be split into 3 subsets. As before, it is a question of the number of ways in which this may be done in a particular case. The argument above applies again.
N-level distinction: Clearly the argument may be generalized for N-level distinctions although, in the light of earlier arguments, N is unlikely to exceed about 10.
Now the procedure adopted to clarify the ordering al any particular N-level, effectively clarifies the nature oi the most fundamental distinction for N = 2, 3, 4 . . . N This in turn provides an ordered configuration of aspects which exemplify the nature of the original totality (i.e N = 1) which was explored by subdivision.
In addition to proceeding by subdivision, clarificatior concerning a named set (e.g. development) may b. sought by determining of what sets it may be considered to be a part. Note that many of the existent fundamen tat sets are identified or named by enumerating their ele meets. The name of the set, if any, derives from them is their plurality and not from any concept of the singular totality they constitute as a set (e.g. human values, human rights, etc.)
2-level combination: The set may, for example, be paired with one other set to form a 2-element set. But in how many ways may this be done in a particular case, given that the pairing cannot be arbitrary but must be based on some aspect of the quality associated with the number 2 (see Annex 2, for example). Such combinations could be ordered and clarified as suggested by the previous section.
3-level combination: The set could be grouped with 2 other sets to form a 3-element set. As before it is a question of ordering the ways in which this may be done to clarify the many possible aspects of the superordinate set.
N-level combination: Again the argument may be generalized, although it is unlikely, as before, that the total in the resulting set would exceed about 10. In this procedure it may well be that particular combinations are not meaningful or useful. Clearly it becomes increasingly difficult, as N increases, to integrate the original set into a combination. But at any stage, a further procedure may be adopted to identify, for an N-level combination, what, successively, the elements of an N-1, N 2....N M combination are. This clarifies the aspects of the nature of the more fundamental superordinate sets (where N-M = 1) which may underly any given set. Again the qualitative characteristics of number (Annex 2) may be used as a guide.
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