1978

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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

Abstract | Part 1-3 | Notes | References | Annex 1 | Annex 2 | Annex 4

**1. Eliciting subordinate sets: relating distinctions**

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.

**2. Eliciting superordinate sets**

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.

Abstract | Part 1-3 | Notes | References | Annex 1 | Annex 2 | Annex 4