2.5 
THE CATENA'S POSITION IN A DERIVATION
SYSTEM 
2.5.1 
BASIC OR ORIGINAL CATENAS AND DIFFERENCE
CATENAS 
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 positivitydifference
in longitude is a socalled 'bicatenal positivitydifference
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,
neutralitydifference 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
neutralitydifference 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 positivitydifference catena its
extremitydirectedness, and in the event of a
neutralitydifference catena its neutralitydirectedness.
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
positivitydifference (the distance) between two longitude catenals
is small, that is, smaller than a certain other positivitydifference
(the neutral or perineutral distance). From the
proximitycatenary 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 moduluscatena".
Such a moduluscatena is a bicatenal monovariant neutralitydifference
catena. The proximity catena is therefore the moduluscatena of the
bicatenal bivariant positivitydifference 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 moduluscatena. Distance is now called
"length". However, the shortness catena is not the moduluscatena
of the bicatenal, but of the monocatenal bivariant
positivitydifference 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
partpredicate would, then, only imply having a certain wholepredicate.
We shall therefore say that difference catenas are:
 bicatenal and monovariant (like all modulus catenas); or
 bicatenal and bivariant (like the original catena of the
proximity catena); or
 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
x_{1} 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 x_{2} 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  = 
{ x_{3};  dx_{2} 
>  dx_{1}  }, 
N  = 
{ x_{3};  dx_{2} 
=  dx_{1}  }, 
M  = 
{ x_{3};  dx_{2} 
<  dx_{1}  }, 
on the condition that dx = dx for d=+INFIN, and
dx = xd for d=INFIN
The definitions for the specific difference catenas are now:
 positivitydifference catena: d.c. for which d=supX
(possibly d=+INFIN);
 neutralitydifference catena: d.c. for which d=0 ;
 monovariant difference catena: d.c. for which
x_{2} is constant
(x_{2}=c) (while x_{1}
is variable);
 bivariant difference catena: d.c. for which both
x_{1} and x_{2} are
variable;
 monocatenal difference catena: d.c. for which it is possible that
x_{3} ¹ x_{1}
and
x_{3} ¹ x_{2}
(catenal 1 and catenal 2 are component parts of catenal
3);
 bicatenal difference catena: d.c. for which necessarily
x_{3}=x_{1} ;
 moduluscatena: bicatenal monovariant
neutralitydifference catena. (This derivation amounts to taking the
modulus of the original catena's value:
P = {x_{1} ; c >
x_{1} } or
P = {x_{1} ;
x_{1}< c} , and so
forth.)
It is worth noting, firstly, that in the general definition
of the difference catena positive difference is positivitymoreness
and evaluated positive by entering > instead of < for the
comparative positivity in  dx_{2}  >
 dx_{1}  ;
and secondly, that the value of d is found back in the name of the kind
of comparative catena  only moduluscatena is a special case.
Thus we speak of "a positivity" and of "a neutralitydifference
catena". It does not even make sense to distinguish a
'polaritydifference catena', as there is no unique aspectual value
which pertains to it. On the other hand, 0 is the aspectual
value of the neutralitydifference 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 neutralitymoreness of the derivative catena. In
many instances traditional thought has been geared to polaritymoreness
catenization with the accompanying negative evaluation
of neutralitymoreness. So slowness may be conceived of as the
negative opposite of fastness which is positive and affirmative
on the polaritymoreness 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.
