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:

  1. according to the range of catena values; this yields 'finite', 'semi-finite' and 'infinite catenas';
  2. according to ordinary language; on the basis of this criterion one can distinguish 'explicit triads', 'quasi-duads', 'quasi-hexaduads' and 'quasi-monads';
  3. according to the position in a derivation system; this categorization starts with a distinction between 'basic' and 'derivative catenas'; and
  4. 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|  |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 temperature.

©MVVM, 41-57 ASWW

Model of Neutral-Inclusivity
Book of Instruments
Catenas of Attributes and Relations
Ways of Classifying Catenas