Logically necessary relationships between catenary variables are of great import. Consider, for example, the case in which one can only have a predicate corresponding to the catena value 0 with respect to the one catena, while having a predicate corresponding to a negative or positive value with respect to another catena. It is then logically impossible to be neutrally catenal with respect to both catenas at the same time.

We shall not deal here with those functional relationships which are logically contingent, such as causality, but only with those which exist on the basis of the definitions of the catenas concerned. To make these relationships explicit we shall express them as mathematical functions with each variable assuming one of the values of one catena's value collection. By means of these functions new catenas are derived, as it were, from other ones. If the derivation does not change the dimension, we shall call the catenas between which the relationship exists "equidimensional to each other".

Catenas which are not derived from others are in our terminology 'basic catenas'; and the catenas derived from them 'derivative catenas'. While all basic catenas are 'original catenas', any derivative catena may itself be an original catena in a further derivation. Together all the catenas that are derived from one and the same collection of equidimensional catenas not derived from one another (the base) form with that base a catenary derivation system. Thus the basic catenas of the spatiotemporal derivation system are those catenas which correspond to one spatial dimension: let us say, the 'longitude', 'latitude' and 'altitude catenas'. (There may be more.) Every primary thing that is catenal with respect to any of these basic catenas is catenal with respect to all of them, and with respect to the whole catenary derivation system.

Of two primary things which are catenal with respect to the longitude catena, the longitude of the one is either more (positive), the same or less (more negative) than the longitude of the other. The catena corresponding to this positivity-difference in longitude is a so-called 'bicatenal positivity-difference catena'. If neither primary thing is fixed or a particular object, the catena concerned is bivariant; if one of them is fixed, then monovariant. (They cannot be fixed both, because then there would only be one value and potential value, whereas a catena requires at least three potential values.) Now, the longitude of the one primary thing is also more neutral, the same or less neutral than the longitude of the other, whatever the neutral longitude may be. Hence, we can also derive bicatenal, monovariant and bivariant, neutrality-difference catenas from basic or other original catenas.

When primary things are considered as wholes in themselves, and when they are then compared, the positivity- or neutrality-difference catena is bicatenal. But the primary things in question may also be component parts of one and the same whole. If the catena value is then a value of this whole, and not of one of its component parts, the difference catena in question will be termed "monocatenal". Given that the two component parts have their own characteristics (like head and tail), monocatenal bivariant difference catenas of the longitude catena indicate the direction of the primary whole concerned: in the event of a positivity-difference catena its extremity-directedness, and in the event of a neutrality-difference catena its neutrality-directedness.

Two longitude catenals may be said to be close to each other, far from each other or something in between. It does, then, not matter whether the one catenal has a more positive, or a more neutral, longitude than the other. So there is also a proximity catena of proximity, farness and a concatenate neutrality or perineutrality. Proximity means that the modulus of the positivity-difference (the distance) between two longitude catenals is small, that is, smaller than a certain other positivity-difference (the neutral or perineutral distance). From the proximity-catenary angle it is irrelevant whether this difference is positive or negative. (It is also then that every 'distance' is nonnegative by definition.) That is why a catena like the proximity catena will be called "a modulus-catena". Such a modulus-catena is a bicatenal monovariant neutrality-difference catena. The proximity catena is therefore the modulus-catena of the bicatenal bivariant positivity-difference catena of the longitude and similar, basic catenas.

For the proximity catena the two longitude catenals in question are considered as separate wholes, but if they are conceived of as those component parts of one and the same whole which differ most in longitude, proximity is nothing else than shortness and farness nothing else than longness (insofar as this one dimension is concerned). Like the proximity catena, the shortness catena is a modulus-catena. Distance is now called "length". However, the shortness catena is not the modulus-catena of the bicatenal, but of the monocatenal bivariant positivity-difference catena of the longitude and similar catenas.

Monocatenal monovariant catenas would relate to the value of a whole with at least two component parts, of which one were to have a fixed predicate. Such catenas do not seem to exist, or if existing, do not seem to play a role, for having a certain part-predicate would, then, only imply having a certain whole-predicate. We shall therefore say that difference catenas are:

  1. bicatenal and monovariant (like all modulus catenas); or
  2. bicatenal and bivariant (like the original catena of the proximity catena); or
  3. monocatenal and bivariant (like the original catena of the shortness catena).

By means of mathematical symbols we can clearly define the several kinds of derivative catenas mentioned here. Let, then, x be the value of the original catena predicate, and X the set of all these values. We use the symbol d (of direction) to indicate the value x in respect of which a greater, equal or lesser deviation of two things is compared. Thus d = 0 when considering what is more neutral, and d > 0 when considering what is more positive. If a thing has a value x1 with respect to an original catena, d determines the kind of catena with regard to which this thing is catenal too. We shall therefore call the catena value d "the aspectual value". A comparative value x2 now determines the kind of catena element the thing in question has with respect to the catena determined by d. This value of comparison may also be a constant c.

If P is the set of derivative catena values corresponding to the positivity, N the singleton of the value corresponding to the neutrality and M the set of values corresponding to the negativity, then a difference catena is in general a catena for which

P= { x3; | d-x2 | > | d-x1 | },
N= { x3; | d-x2 | = | d-x1 | },
M= { x3; | d-x2 | < | d-x1 | },
on the condition that |d-x| = d-x for d=+INFIN, and |d-x| = x-d for d=-INFIN

The definitions for the specific difference catenas are now:

  1. positivity-difference catena: d.c. for which d=supX (possibly d=+INFIN);
  2. neutrality-difference catena: d.c. for which d=0 ;
  3. monovariant difference catena: d.c. for which x2 is constant (x2=c) (while x1 is variable);
  4. bivariant difference catena: d.c. for which both x1 and x2 are variable;
  5. monocatenal difference catena: d.c. for which it is possible that x3  x1 and x3  x2 (catenal 1 and catenal 2 are component parts of catenal 3);
  6. bicatenal difference catena: d.c. for which necessarily x3=x1 ;
  7. modulus-catena: bicatenal monovariant neutrality-difference catena. (This derivation amounts to taking the modulus of the original catena's value: P = {x1 ; |c| > |x1| } or P = {x1 ; |x1|< |c|} , and so forth.)

It is worth noting, firstly, that in the general definition of the difference catena positive difference is positivity-moreness and evaluated positive by entering > instead of < for the comparative positivity in | d-x2 | > | d-x1 | ; and secondly, that the value of d is found back in the name of the kind of comparative catena -- only modulus-catena is a special case. Thus we speak of "a positivity-" and of "a neutrality-difference catena". It does not even make sense to distinguish a 'polarity-difference catena', as there is no unique aspectual value which pertains to it. On the other hand, 0 is the aspectual value of the neutrality-difference catena which comprises exactly the same predicates. (Note that comparative catenas with a nonextreme, nonneutral aspectual value are not recognized in ordinary, nontechnical language.)

The perineutrality of an original catena has to be evaluated positive as the neutrality-moreness of the derivative catena. In many instances traditional thought has been geared to polarity-moreness catenization with the accompanying negative evaluation of neutrality-moreness. So slowness may be conceived of as the negative opposite of fastness which is positive and affirmative on the polarity-moreness approach. Yet, in the theory of catenas we shall stick to the positivity of moreness and the crucial role of the aspectual value in the naming of comparative catenas and their predicates.

©MVVM, 41-57 ASWW

Model of Neutral-Inclusivity
Book of Instruments
Catenas of Attributes and Relations
The Catena's Position in a Derivation System