THE TRIPARTITE STRUCTURE OF THE
The concept of an attribute- or relation-catena looks
very much like that of a dimension, scale, spectrum or factor.
Yet, while there is much conformity, there are sound reasons to
use the term catena where the concept of dimension, or a
similar concept, would not be clear enough.
One reason, of course, is that the catena is a thing, albeit abstract,
with its own office in our ontological edifice: it is not a set of values
or something of that ilk, which would further force us to accept not only
the existence of sets but of values as well.
Another reason is that a number of different catenas may have the same
And a third reason is that the components of a catena cannot
only be ordered linearly but that they belong, and must belong,
to one of three typical classes by definition.
What are the characteristic classes or subsets making up the
extensionality of a catena?
Let us have a look: to the catena of electropositivity and -negativity
also belongs electroneutrality; to that of happiness and unhappiness also
being neither happy nor unhappy (but sentient nevertheless); to that
of more and less also equally; to that of betterment
and worsening also the continuation of the same goodness, badness or state
of being-neutrally-indifferent; to that of acid and basic
also neutral (in the sense of neutrally neither acid nor
basic); to that of cold and heat also the normal or moderate
temperature corresponding with being neither (too) cold nor (too) hot; to
that of weakness and strongness also being neither weak nor strong (but
something in between); and so on.
As to rest and motion, we can subdivide motion into motion in positive and
in negative direction (for each dimension concerned); as to
continuation and change, change can be divided into decrease
and increase; as to indifference and difference or not being
indifferent, the latter one is positive (more or, for
example, goodness, happiness or a liking for something) or
negative (less or, for example, badness, unhappiness or a
dislike of something); as to normality and abnormality, abnormality
is positive or negative (for the factor concerned, if
normal is taken in a statistical, or similar, sense); as to
being-balanced and -unbalanced, the latter is a question of too
large an amount or too small an amount; and so on and so forth.
Electropositivity and -negativity, acidness and being-basic,
happiness and unhappiness admit of degrees. Similarly, there are
different intensities of weakness and strongness or strength, of
the positive and negative forms of not being indifferent or
difference, of friendliness and unfriendliness; and so on. There
are therefore more than one, maybe many, electropositivity- and
-negativity-predicates, happiness- and unhappiness-predicates,
weakness- and strength-predicates, difference predicates, and so
on and so forth.
For each degree, for each intensity, there is a distinct
(secondary) attribute, and a proper
primary predicate can have only one of
these attributes at a time.
Heaviness, for instance, may denote each heaviness attribute
separately if thought of as a proper attribute; it may also refer to all
heaviness attributes together — be their common denominator, as
It is each heaviness attribute separately, however,
which is an extensional element of the heaviness catena, not the
set or totality of all heaviness attributes. The conceptual set
of all heaviness attributes is a subset of the catena; and it is
a 'positive' subset in that all its members have the secondary
attribute of positivity, not in that it has this attribute
itself (as it simply is not a thing with attributes as intensional
elements). It follows that also neutral attributes like
electroneutrality, the neutrality of the acidness catena or neither
heavy nor light (but having a weight nevertheless) are
elements of the catena themselves, and not their singleton, or
some whole of which they would be the sole component part.
It should be evident now that no fewer than three sorts of
catenated attributes or relations belong to each
catena, namely one or more positive predicates, one neutral
predicate (with the catena value 0) and one or more negative
Altho the number of
extensional catena elements is not fixed, the extensionality of the catena
has always three subsets (which are conceptual constructs however): (1)
the subset of positive predicates, (2) the singleton of the neutral
predicate, and (3) the subset of negative predicates.
The predicate with the
catena value 0 we shall call "a limit element" because
it is a limit between the positive predicates on the one hand and the
negative ones on the other.
(Later, we will see that it is not necessarily the physical, chemical or
other quantity which is 0 in the case of neutrality.)
In a way neutrality is a limiting case of both positivity and negativity,
the point where positivity and negativity could be said to overlap or
On the basis of its tripartite structure we shall define the
"a whole of catenated primary predicates of which
the extensionality can be divided into three
subsets so that the only element of one of these subsets is the limit
element between the mutually opposite elements of the
other two subsets".
(Note that the requirement is that the extensionality can be divided in
In theory and in
e it may be
divided in one or more other ways as well.)
The neutrality of a
catena must be a limit element between the subset of positivities
and that of negativities. Electroneutrality, for instance,
may be neutral and a limit element, it is not a limit element
between happiness and unhappiness, and therefore happiness,
electroneutrality and unhappiness do not constitute a catena.
Similarly, altho electropositivity is a positive predicate and
unhappiness — let us assume — a negative one, they still do
not have a common, neutral limit element.
So they are not each other's opposite and do not belong to the same
catena: they are both catenated but not concatenated.
Sets of extensional elements of one catena are closely
related to sets of numbers. There is a one-to-one correspondence
between catenated elements and the numbers associated with the
degree of actualization. When we speak of "opposite predicates",
however, we use the term opposite in a broader sense than a
mathematician speaking about numbers might do. A mathematician
may solely speak of "oppositeness" when the absolute values are
equal. Only -1 would thus be opposite to +1, and not -2 or any
other negative number. In the theory of catenas this conception
would be too narrow for then happiness and unhappiness, for
instance, could only be called "opposites of each other" if the
respective intensities or degrees of actualization happened
to be equal. But happiness and unhappiness are opposites
regardless of the degrees involved; and so are love and hate,
increase and decrease, highness and lowness, and so on. (Also
etymologically it is justifiable to use opposite in this
general way.) In accordance with this we shall call all those
(catena) values "opposites" which differ in (plus or minus)
sign. Hence, all negative numbers are 'opposite' to +1, and all
positive numbers are 'opposite' to -0.5 or any other negative
While referring to the corresponding set of degrees of
actualization (interpreted as numbers) we may now say that a
predicate is an opposite of or opposite to another predicate,
(1) if its degree of actualization is opposite to that of the
other predicate, and (2) if they belong to two subsets with a
common, neutral limit element, or if they are elements of the
same catena (the catena itself having already been established).
And just as we may, loosely, call the subset of concatenate
predicates itself "positive" or "negative", so we may call these
subsets "opposites" as well, altho strictly speaking, it is only
the predicative elements which are opposites. Thus deceleration,
for instance, is the opposite of acceleration and a deceleration
of -1 m/sec2 is opposite to all acceleration predicates. If the
deceleration is -1m/sec2, this is its degree of actualization:
negative in the case of deceleration, positive in the case of
acceleration. The common limit predicate catenated to both
deceleration (all deceleration predicates) and acceleration (all
acceleration predicates) is the continuation of the same velocity.
Since a proper predicate corresponds to one degree of
actualization only, an object with an acceleration of +1m/sec2
has nothing in common with an object with an acceleration of
+2m/sec2 (as far as this aspect is concerned). Only if they both
have exactly the same acceleration (positive, 0 or negative), do
they have a predicate of the acceleration catena in common. What
is meant by saying that two objects 'have their (positive)
acceleration in common' is that they both have a predicate
belonging to the positive subset of the acceleration catena.
It is, then, not the objects but their predicates which
have something in common, namely the secondary predicate of
Expressions such as deceleration, weak and irrelevant
are called "marked terms" because they are employed in only one,
restricted sense, whereas the corresponding (but not necessarily
opposite) 'unmarked terms' (acceleration, strong,
relevant and the like) have both a specific meaning and a general
meaning relating to the whole dimension in question.
In terms of the catena (where applicable) marked terms refer to a proper
subset of its extensionality (in the simplest case the negative
subset) and unmarked terms either to a proper subset (for
example, the opposite, positive subset) or the improper subset
of the total extensionality itself.
Hence, in what is its acceleration? acceleration may either
be positive and the opposite of deceleration or it may mean acceleration catena
predicate whether the predicate is positive, neutral or negative.
Sometimes there is a marked term in addition to the
unmarked one to refer to positive predicates. For example,
longness and strongness can only refer to the opposites of
shortness and weakness. But length and strength are unmarked
and even things which are short and weak have a length and a
strength, just as things which are light have a weight. To have
a length, then, means to have one of the predicates of the
length catena. Length in this sense does not denote a
catenated predicate, but stands for the total spectrum ranging
from extreme shortness to extreme longness. The occurrence of
ambiguous, unmarked terms is one of the deficiencies of the
ordinary variant of the present and many other languages.
Usually one has to put up with a term which may denote either the
positivities or other predicates of a catena, which will distinguish them
from its negativities or other concatenated predicate(s), or the
extensional predicates of a catena in general, which will distinguish them
from the extensional elements of all other catenas.