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 those 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 supplement of each other. Two catena (extensionality) subsets are the catena (extensionality) supplement 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 like 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.