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 0-point 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 û1 is the catena value of the original, first catena, and û2 that of the derivative, second catena, then it is the derivation function which describes the relation between these two catena values: û2= d (û1).

Instead of "û2= d(û1)" we may write "û2 = e(û1) + E". We now define a factitious derivation as a derivation for which (necessarily) û10 and/or E0 if û2=0. For a basic catena E=0 and û2 = û1, and therefore û1=0 if û2 = 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 û2= A × ðû1÷ðw + B = A×f ' (û1) + B (for positivity-differential catenas A×f+1) + B, for neutrality-differential catenas A×f01) + B; A0). Here û2 = 0 does not in any way determine û1, nor B. Hence, û1 is not necessarily positive or negative if û2=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 zero-level of reiteration there is a clear difference between necessarily and not necessarily factitiously derived comparative catenas. Thus bicatenal monovariant positivity-moreness- or -increase-catenas are nonfactitiously derived if B=0 in û2=Aû1 + B with A0. (If A=1, then the nonfactitiously derived positivity-moreness catena is identical to the original catena.) If B0, then we must consider them factitiously derived.

The formula of the derivation of a bicatenal monovariant neutrality-moreness- or -increase-catena is û2 = A | û1 | + B (A0 and B÷A<0). Thus B0 and also û10 if û2=0. Hence, all modulus-catenas are derived in a factitious fashion. Bivariant comparative catenas, on the other hand, are nonfactitious if one takes E=0, because û2=0 determines neither E in û2 = e (û1,11,2) + E nor û1,1 or û1,2.

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 û1 if û2=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 neutrality-moreness 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.

©MVVM, 41-68 ASWW

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