2.2.2

THE TRIPARTITE STRUCTURE OF THE CATENA

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 the existence of sets and of values. Another reason is that a number of different catenas may have the same dimension. 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 (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 those 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 were. 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.

Now it is evident 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 predicates. 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. (We will see later 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 meet.

On the basis of its tripartite structure we shall define the catena as

"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".

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 number.

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 positivity.

Expressions like 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. Thus 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 as opposed to its negativities or other concatenated predicate(s), or the extensional predicates of the catena in contradistinction to those of all other catenas.