2.2.3 |
SUBDIVISIONS OF THE CATENA'S EXTENSIONALITY |
In the theory of catenas attributes and relations are characterized on the
basis of their position in, or with respect to, a
catena or system of catenas. We shall say that
this theory and any other body of thought which rests on its conceptual
framework is of a 'catenical' nature.
It is not just the reference to
catenated predicates and other predicates related
to the concept of the catena which is typical of catenical thought
--thought is practically unthinkable without
reference to such predicates-- but also the
recognition of these predicates as belonging or related to one or more
predicate catenas.
We shall draw a distinction between a predicate's being
'catenated' and its being 'catenary'.
A catenated predicate is a predicate of a catena as extensional
element, that is, a component proper part of a catena.
A catenary predicate, on the other hand, is a predicate of a catena
as intensional element or a predicate which the catena, or one of its
parts, solely has as a catena, or as a part of a catena.
The meaning of this catenary is being or pertaining to a
catena.
So the catena itself is a catenary thing (and not a catenated thing).
And if two predicates are opposed or otherwise catenated to each
other, there is a catenary relation of oppositeness or catenatedness
between them. The neutralness (or 'neutrality') of
neutral, catenated predicates is a catenary attribute because
they can only be neutral (in the catenical sense) as part of a
catena. Catenated predicates are first-order, and catenary
predicates such as oppositeness and neutralness second-order
predicates. Like all predicates they may be proper or improper.
Given a certain collection of attributes or relations which is the
extensionality of a catena, there are
theoretically countless ways of subdividing such a collection. (There
are already four possibilities if there is merely one negative and
merely one positive predicate besides the neutral one.)
In
practise however, that
is, in ordinary language, only a limited number of types of subdivisions
are found and do make sense.
The first subset to be distinguished, then, is the
singleton which solely contains the extensional catena element
with the catenical value 0 as degree of actualization between
negative values on the one hand and positive ones on the
other. This neutral predicate on the line we have termed "a
neutrality" (but not "a neutralness").
In the first instance
tho, neutrality is a
catenary, secondary attribute (and with this meaning of neutralness
an expression like the neutrality of the catena has no reference).
Yet, by also calling a neutral
thing "a neutrality" we make use of a feature of the present
language in which things which have a certain predicate often
acquire the name of this predicate itself, preceded by the
definite or indefinite article. For example, 'a special(i)ty' is
something special, 'a beauty' is something or somebody beautiful,
'an abrasion' is something which has been abraded. (Compare
also fellowship, community, externality or other
multiplicities of meaning.)
In a loose, metonymical sense the singleton of the neutral
catena predicate may be said to be 'neutral' too, and to be 'a
neutrality' (altho the set itself does not even exist). All
the other predicates, and catena subsets of predicates which
are not neutral, we shall term "unneutral" or "polar", and
"unneutralities" or "polarities". Noncatenated predicates
are neither neutral nor unneutral, but nonneutral and
non-unneutral. Even a catenary predicate such as neutrality itself is,
strictly speaking, both nonneutral and non-unneutral. (Un-
negates the meaning of the base word within the next higher
framework which is that of the catena in the case of neutrality;
non- negates in general, subject to no restriction.)
A catena element or subset corresponding to positive catena
values only, or to negative catena values only, will be termed
"monopolar", or "a monopolarity". In the former case it is a
positivity, in the latter case a negativity. Catenically,
positivity has nothing to do with certainty, actuality,
affirmation or agreeableness, and negativity nothing
with negatoriness or disagreeableness (at least not on the
basis of its being positive rather than negative, or its being
negative rather than positive).
It is of paramount importance here to guard against traditional
misassociations: the catenical definitions of positive and
negative are based on mathematical and similar usage, and a positive
number is not certain or uncertain at all, nor affirmative, while a
negative number is not negatory at all.
Negative is as much nonpositive as
positive is nonnegative and we can be as sure that something
is negative as we can be that something else is positive, or neutral for
that matter.
The neutrality of a catena is the limit element closed in by
the catena's positivity on one side and by the catena's
negativity on the other. In general however, any predicate or
predicative subset 'limits' another catenated subset, if it
corresponds to only one value, and if this value is one of the
two values (theoretically among them possibly +INFIN and -INFIN)
between which the values of the subset limited by it lie
(assuming that all interjacent catena values belong to the value
set of the predicative subset). Thus, altho the neutrality (subset)
is a limiting unit, for it limits the positivity (subset)
and the negativity (subset), this does not mean that, conversely,
a limit predicate or subset must be neutral too. On the
contrary: extremities (which are extremely unneutral) also limit
positivities and negativities but then at the other end of the
scale.
A predicative subset is 'limited by' another one, if the
latter limits the former. The positivity and the negativity of a
catena are both limited by the catenated neutrality, that is,
the neutral predicate of the same catena. The duad of positive,
complete monopolarity and negative, complete monopolarity may
be said to be limited by the neutrality as well, if we
only look at the corresponding absolute values. This duad we
shall call "the complete bipolarity" of the catena. Both
monopolarities and bipolarities, whether complete or not, are
unneutralities for they do not have the value 0 in the corresponding
set of catena values. Complete bipolarity is unneutrality,
and also bipolarity, in the broadest sense. (Complete is used
here in the sense of comprising all extensional catena elements
with the same, proper or improper, catenary predicate.)
A neutrality and a complete bipolarity which is catenated to
it (and thus limited by it) are the catena supplements of each
other. Two catena (extensionality) subsets are the catena
(extensionality) supplements of each other, if the union of both
subsets is a catena extensionality. Two catena subsets which are
opposites of each other cannot be catena supplements of each
other (as they do not include the neutrality), and two catena
subsets which are supplements of each other cannot be opposites
of each other (as the neutrality which is one of them is not
opposed to the predicates of the other.) Electropositivity and
electronegativity, acid and basic, intelligence and
unintelligence are opposites and not catena supplements of each
other. On the other hand, rest and motion, equality and
difference, normality and abnormality (in a statistical or
similar sense) are supplements and not opposites of each other.
The traditional confusion of the distinction between opposition and
supplementation --and several other types of
relations between
primary predicates and
things-- is closely related to the disregard for
the neutrality and for the total catenary structure it belongs to.
All proper, catenated predicates are 'atomic' in that they
correspond to only one value or degree of actualization, and in
that they are thus an atom in the primary world. To 'have a
certain proper, catenated predicate' is to have it in the most
literal sense. When we speak about "the positivity" and "the
negativity" of a catena, however, we refer to subsets of usually
several positive and several negative predicates and their
corresponding, positive and negative values. This is why the
positivity and negativity are improper predicates of the catena
(unless there happens to be only one positive and only one
negative degree of actualization). To have the/a positivity or
negativity of a catena is an abbreviation for saying that the
primary thing in question has one of the proper, catenated
predicates which belong to the positive or negative subset of
this catena. (Note that to be happy, for instance, means to
have the/a positivity of the happiness catena and is an
abbreviation or 'generalization' of the same kind.)
Neutralities are always atomic and proper but unneutralities
may be either proper or improper predicates. Improper
unneutralities can be subdivided into bipolarities and improper
monopolarities (either complete or noncomplete), dependent on
whether they are both positive and negative, or only positive or
negative. The sole catenated, proper nonpolarity is the neutrality;
the most important kind of catenated, improper nonpolarity
is the so-called 'perineutrality'. This is a predicate
of which the catenical value collection contains only low
positive values, the catena value 0 and high negative values
(close to 0). To express perineutrality, that is, 'moderateness'
or 'moderation', people often employ the terms medium or
middling, sometimes without even mentioning the positivity of
the catena (as in large, medium or small and it's middling).
Medium large, of middling height and moderately
expensive are all perineutral, improper nonpolarities. (In agreement
with the use of positive, neutral and so on, we shall use
the term perineutral for the predicate itself and not for the
persons or other primary things which have this predicate.) The
reference to perineutralness rather than to neutralness is
typical of catenas of so-called 'vague' predicates, that is,
catenas for which it is not immediately clear which predicate is
the neutral one (while the collection of physical or other
noncatenical values one deals with in the first place also contains
only nonnegative values, with or without 0 as an extreme
noncatenical value).
In everyday language many more kinds of catenical concepts
are found besides the perineutral moderate(ness) and the more
simple ones. Every gradation in the language with respect to an
(improper) catenated, primary predicate concerns a subdivision
of a catena extensionality. So the concept very intelligent,
for instance, is the concept of a special kind of noncomplete,
improper monopolarity (corresponding to high positive catena values).
The attribute (or derelativized relation) being-the-most-intelligent
is a special kind of atomic polarity, namely an
'extremity'.
Predicates such as moving fast and a little asymmetrical are
noncomplete bipolarities.
The connected series of predicates or predicative subsets which are
extremely positive, quite positive, positively perineutral, neutral,
negatively perineutral, quite negative, very negative and extremely
negative could be presented as a prototype of the predicate catena as much
as the trinity of (complete) positivity, neutrality and (complete)
negativity.
Some typical subdivisions of the catenary extensionality are shown in
figure I.2.2.3.1.
