2.5.3 
FACTITIOUS AND NONFACTITIOUS DERIVATIONS 
A catenical value belonging to a certain catena may
be, but need not be, equal to some empirical value relating to
this catena. It may be that the choice of empirical quantity is
arbitrary, both with respect to the 0point selected and with
respect to the unit used. Thus the empirical value may be always
positive or nonnegative, in which case it is impossible that it
would be equal to the catenical value. In general there is a
relationship between the catenical and empirical value which can
be expressed as û=k(v). In this formula û [v with a
caret over it] is the catena value, v the empirical value and k
what we shall call "the catenization function".
Whereas derivation is a transformation from the one catena
or form of catenality to the other catena or form of catenality,
catenization is a transformation from what is possibly not
clearly catenary to what is explicitly catenary. Now, if
û_{1} is
the catena value of the original, first catena, and
û_{2} that of
the derivative, second catena, then it is the derivation
function which describes the relation between these two catena
values: û_{2}= d
(û_{1}).
Instead of "û_{2}=
d(û_{1})" we may write
"û_{2} =
e(û_{1}) + E". We now
define a factitious derivation as a derivation for which (necessarily)
û_{1}¹0 and/or
E¹0 if û_{2}=0.
For a basic catena E=0 and û_{2} =
û_{1}, and therefore
û_{1}=0 if û_{2} =
0. Hence, basic catenas are not factitiously derived, still regardless of
the fact that they are, properly speaking, not derived at all. The
codification of catenas according to the factitiousness of their
derivation is thus really another subdivision of derivative
catenas. It is a codification in addition to that according to
the operational level of reiteration.
For differential catenas û_{2}= A ×
ðû_{1}÷ðw + B =
A×f ' (û_{1}) + B (for
positivitydifferential catenas
A×f^{+}(û_{1}) + B,
for neutralitydifferential catenas
A×f^{0}(û_{1}) + B;
A¹0). Here û_{2} = 0
does not in any way determine û_{1}, nor B.
Hence, û_{1} is not necessarily positive or
negative if û_{2}=0, and therefore
differential catenas are nonfactitious catenas if B=0. The derivation is only
factitious if B is taken to be positive or negative.
On the zerolevel of reiteration there is a clear difference
between necessarily and not necessarily factitiously derived
comparative catenas. Thus bicatenal monovariant positivitymoreness
or increasecatenas are nonfactitiously derived if B=0 in
û_{2}=Aû_{1} + B
with A¹0. (If A=1, then the nonfactitiously
derived positivitymoreness catena is identical to the original catena.)
If B¹0, then we must consider them factitiously
derived.
The formula of the derivation of a bicatenal monovariant
neutralitymoreness or increasecatena is
û_{2} = A
 û_{1}  + B
(A¹0 and B÷A<0).
Thus B¹0 and also
û_{1}¹0 if
û_{2}=0. Hence, all moduluscatenas are
derived in a factitious fashion. Bivariant
comparative catenas, on the other hand, are nonfactitious
if one takes E=0, because û_{2}=0 determines
neither E in û_{2} =
e (û_{1,1},û_{1,2})
+ E nor û_{1,1} or û_{1,2}.
The reason to distinguish factitious from nonfactitious
derivations is that this creates additional clarity with respect
to the relationship between physical 'reality' (or theory) and
catenical theory. The recognition that a predicate's or catena's
transformation is artificial is of importance where the exact
assessment of the neutral, empirical value is concerned. The
terminology underlines that the choice of a special, polar value
of the value collection of the original catena as the new
'neutral' catena value, and the assignment of some positive or
negative value to the constant E, is farfetched if such
derivative 'neutralness' is meant to be of universal significance.
The choice of any value other than 0 for û_{1}
if û_{2}=0,
or for the constant, is then purely arbitrary. (Where such a
choice is not arbitrary we are dealing with special sets of
catenals in closed systems, something we shall discuss in the
next division of this chapter.)
The variety of possible, factitious and nonfactitious, transformations
could lead to a situation in which different empirical conditions
would correspond to the same sort of catenated
'neutrality'. But because of the central role of neutrality in
the catenary structure it is especially ambiguity with regard to
this predicate which should be avoided as much as possible.
Interaction between different catenas derived from one and the
same original catena can give rise to incompatible conceptions
in which the neutrality of the original catena, or the one
derivative catena, is the polarity of the derivative, or
another derivative, catena. Such confusion does not exist where
the original catena's neutrality remains a neutrality in the
derivation, and it need not exist where the derivation itself
generates a new empirical perspective (as in the case of
differential catenas) or where the catenals of the original
catena are not the catenal of the derivative catena (as in the
case of monocatenally derived difference catenas).
Just as there is a rule in mathematics that evolution goes
before multiplication  2×3××2=18, and not 36
and just as one can make known one's deviation from this rule
by means of additional symbols (2×3)××2=36,
so we shall for the sake of
clarity adopt the rule that nonfactitious goes before factitious
with regard to catenary derivations. This rule of nonfactitious
priority reads in full: "unless it is mentioned explicitly, or
is implicit in its definition, that a predicate or a catena to
which it belongs has been transformed factitiously, this predicate
or catena is taken to be derived in a nonfactitious way".
This rule does not affect the sign of the monopolarities; and
rightly so. Whether a monopolarity is evaluated positive or
negative, and its opposite the other way around, is of no import
so far as the catenary structure is concerned.
According to the rule of nonfactitious priority predicates
such as normality and abnormality are each other's catena
supplement and not opposite. By assuming their relationship to
be one of catena supplementation they constitute a nonfactitious
catena, namely the abnormality catena of which normality is the
neutrality. This means that something that or someone who
deviates in only the slightest (recognizable) degree from the
mean or 'mode' is, strictly speaking, already 'abnormal'. If we
considered the relationship between normality and abnormality to
be one of opposition, these opposites would constitute the
normality catena, but this catena is a necessarily factitious,
comparative catena (with the abnormality catena as an original
catena and one degree of original abnormality as the arbitrary,
new limit of derivative 'normality'). Should one speak explicitly
of a normality catena, or the neutralitymoreness
catena of the abnormality catena, normality (as a positivity)
and abnormality (as a negativity) will be opposites.
For predicates like slowness and fastness the factitiousness
of their derivation follows already from the meaning of slow
and fast (since the type of comparative catena to which they
belong is always factitious). Thus whatever speed we take as
high (fast) or low (slow) is arbitrary from a universal point of
view. But let us assume that in a certain context 10 km/h is
'slow'. Is then an object which moves in a negative direction at
a rate of 10 km/h negatively or positively catenal? Without
further qualification the answer is that it is negatively catenal.
Only with respect to the slowness catena if mentioned
explicitly is it positively catenal. And without further
qualification it is neutrally catenal if it is at rest, however
'extreme' its slowness may be in this case. Such is of course
not to play down the fact that motion and rest are
necessarily relative concepts themselves, given the absence of
any absolute and universal, spatial frame of reference.
