WAYS OF CLASSIFYING CATENAS
THE FOUR MAIN CRITERIONS OF CLASSIFICATION
When we say that a predicate like larger (than) is
comparative, it is, properly speaking, not correct to suggest
that the predicate by itself has a comparative character. It is
rather the catena to which larger belongs which is comparative.
Smaller and as large as, the predicates catenated to
larger, are just as comparative. It is impossible that a
comparative predicate would be catenated to one which is not,
simply because it is first of all the catena to which it belongs
which is comparative. This illustrates, too, how the nature of a
catena determines the nature of its extensional elements. So the
classification of catenas is indirectly also a classification of
catenated predicates and catenical aspects. Catenated, primary
predicates do not only have secondary attributes or relations on
account of their position in a catena, they also have such
attributes and relations, albeit improperly, on account of the
position the catena they belong to, or refer to, has itself.
Catenas can be categorized in at least four different ways.
The following criterions of classification are partially dependent,
partially independent of one another:
- according to the range of catena values; this
yields 'finite', 'semi-finite' and 'infinite catenas';
- according to ordinary language; on the basis of
this criterion one can distinguish 'explicit triads', 'quasi-duads',
'quasi-hexaduads' and 'quasi-monads';
- according to the position in a derivation system;
this categorization starts with a distinction between 'basic' and
'derivative catenas'; and
- according to the scope of catenization; this
criterion differentiates 'catenas of universal' and 'of special scope'.
We shall treat of the second classification in the following
sections of this division. The third and fourth classifications
are important enough to devote a separate division to.
The first classification that relates to the range of catena
values is mainly of empirical interest. Until now we have taken
it for granted that a complete positivity would comprise all
positive values, and a complete negativity all negative values.
This is, strictly speaking, not correct. What we should say is
that they comprise all the positive or all the negative catena
values, for we may not be justified in assuming that there is a
predicate corresponding to every value or number which is
mathematically (that is, theoretically) conceivable. It may be
implausible in particular to assume that the total set of catena
values is a continuum of which each mathematical value between
two different values corresponding to a catenated predicate
corresponds to a catenated predicate itself as well. Nevertheless,
such a hypothesis would not amount to more than the
acceptance of an infinite number of abstract entities, something
that may not be as questionable as the belief in infinite
collections of concrete objects.
A 'finite catena' is now a catena of which the value
collection has both a lower limit (inf C) and an upper limit
(sup C). The degree of catenality with respect to such a
catena is confined to a minimum extreme value and a maximum one.
A catena is symmetrically finite if, in terms of catena values,
the modulus of the lower limit equals the (modulus of the) upper
limit. (Since there is at least one negative catena value, the
minimum must always be negative, and since there is at least one
positive value, the maximum must always be positive.) If they
are not equal, the catena is asymmetrically finite:
|inf C| ¹ |sup C|.
A catena such as the motion catena is a symmetrically finite
catena as the velocity of light is the maximum velocity
possible, both in a negative and in a positive direction of the
same dimension. This implies that the slowness catena is finite
too, but whether it is symmetrically finite depends on the
position of the neutral value and the way catena values relate,
or are made to relate, to the different velocities; on the
'catenization' so to say.
When applying the same classificatory criterion to the
catena's range of values, we can also distinguish 'semifinite'
and 'infinite catenas'. A semifinite catena is, then, limited
at one side and unlimited at the other, whereas an infinite
catena has neither a lower nor an upper limit. If temperature,
for instance, is conceived of as a physical phenomenon, and not
as something felt by living or sentient beings, the physical
heat catena corresponding to this quantity is semifinite because
there is a lowest temperature and (presumably) no highest