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© Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.

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1. First Order Hierarchies.

2. Second Order Hierarchies.

3. Extreme Cases.

1. First Order Hierarchies.

a. Reconsider the definition: if all X are Y, then “X” is a class of Y (or member of “Y-classes”). The condition only implies that some Y are X.

In the case where all Y are X, they are coextensive and their relation is reciprocal; then “Y” is also a class of X (or member of “X-classes”), and “X” and “Y” are members of each other’s group of classes (which does not mean that they are members of each other, note well); such classes may be called equal. “X” is an equal-class of “Y”, signifies that X-ness and Y-ness are two ‘aspects’ of the same ultimate referents.

But in the case where some Y are not X, they cover a different extension and their relation is uneven. “X” is a member of Y-classes, but “Y” is not a member of X-classes. In such case we say that “X” is a subclass of “Y” and that “Y” is an overclass of “X”. Alternatively, we say that “X” is a lower class than “Y”, and “Y” is a higher class than “X”; or again, we speak of species and genus.

Note in passing, we often define a species by stating its genus (or one of its genera) together with a differentia; the latter is that character in the ultimate referents of a species, which distinguishes them from the ultimate referents of other species of same genus; the referents of all the species have in common the generic character.

Thus, we here introduce three new copulas, one of which is reversible, and two of which are relative to each other. These of course may be denied, making six altogether. These copulas differ from those previously defined, in that the subject and predicate are both nominal. Their function is to establish, or more precisely express, the hierarchies among classes. These various relations have their own logic, which can be analyzed in detail as we did for previous ones; I will not get into that here, however (the reader is invited to do the job).

b. We call division, listing the subclasses of a class; If the subclasses of the latter are in turn listed, we call the process ‘subdivision’. We represent these relations on paper by means of (upside down) ‘trees’, in which the highest class (or summum genus) is placed at the top, and successively divided into lower classes, like a downward branching.

Since all classes ultimately fall under the heading of “things”, there is only one big tree; however, we may speak of branch systems as trees, too. Note that we must have at least one general positive proposition ‘all X are Y’ and/or ‘all Y are X’, to be able to say that “X” and “Y” are in the same tree, or branch of a tree. Otherwise, they are neither equal, nor lower, nor higher classes, in relation to each other, but are in separate trees, or branches of a tree.

If we stand back and consider all possible classes, we see that, though they form a single tree, it is not flat. We have a multitude of hierarchies, all stemming down from “things”, in three dimensions. Hierarchies with entirely different referents, have no intersecting branch lines; hierarchies with some but not all referents in common, have some intersecting branch lines; hierarchies with all the same referents have the same branch lines.

The latter occurs when we have two sets of equal classes: they run along the same branch lines, but they signify different ‘principles of division’, different aspects of the same referents. Thus, for example, humans can be divided into those with male sex-organs and those with female sex-organs, or alternatively, into those without bosoms and those with bosoms: yet these two divisions overlap exactly (ignoring exceptions).

The ultimate referents of all these classes are at the very bottom, in a ‘horizontal’ plane (representing the space-time continuum). There is, as it were, a fanning-out below the lowest classes, to cover the ultimate referents. The relation of referents to lower or higher classes is the same (membership), but it is not the same as the relation of lower classes to higher classes (hierarchy), note well.

2. Second Order Hierarchies.

a. With all this in mind, we see that what a class of classes does is refer us to all the subclasses of a class, plus the class itself. Thus, we should not confuse a class of classes with a first-order overclass, which stands higher up in the continuum of classes.

Whereas an upper first-order class is nominal, and bears certain hierarchical relations to others like it — a class of classes subsumes a class and its subclasses, without thereby becoming part of the same hierarchy, and thus constitutes a second order. Thus, ‘hierarchy’ and ‘order’ are two distinct ways we can stratify classes, and should not be confused.

The two orders of class, “X” and “X-classes”, for any X, are not comparable. The former refers to all things which are X as its members, the latter refers to all (mental) groupings of things which are X as its members. The one concerns numerous individual things, the other untold collectives (in every which way) of these very same things. Their world of reference is one and the same in size, so it is hard to say which is ‘bigger’. The number of referents each has is different, but (in most cases) incalculable.

b. If we apply the definition of classes of classes to classes of classes, we obtain the following result: if all X-classes are Y-classes, then “X-classes” is a class of Y-classes, or a member of “classes of Y-classes”. Here, now, we have classes of classes of classes. We can repeat the process, and obtain an infinity of levels upon levels. But it does not seem to mean anything more than “Y-classes”, to me at least.

The basis on which we form various classes about anything, is in the things they concern. For example, the different kinds of dogs differ in sizes, colors, and so on. Beyond that, the ‘containers’ as such are uniform, there is nothing to distinguish them from each other, other than the differences observed in their ‘contents’. Thus, to pile up level upon level, over and above classes of things and classes of classes of things, is a meaningless redundancy. We may reasonably conclude that there is no order of classification above the two already considered.

c. We may, however, organize second order classes into hierarchies among themselves, on the basis of statements like ‘all X-classes are Y-classes’. In that case, “X-classes” is an equal-class or subclass of “Y-classes”; and similarly in other cases, in accord with the above definitions of hierarchical relations.

Obviously, the hierarchies in the second order reflect those in the first order, on the basis of inferences like ‘if all X are Y, then all X-classes are Y-classes’. This just signifies that formal eductions are feasible from one system to the other.

However, the relationship of second-order to first-order classes is not hierarchical, but simply subsumptive. It is like the relation of first-order classes to their ultimate referents — namely, a relationship of inclusion as members; it is not like the relation of higher classes to lower classes of one and the same order.

For first-order classes, as we pointed out, the theater of reference is the space-time continuum, represented as a horizontal plane. For second-order classes, the theater of reference is the vertical dimension in which the tree of first-order classes evolves. However, the tree of second order classes need not be viewed as implying yet another dimension; we can view it as a distinct branch system within the same vertical dimension. The two orders of classes are layered in neat harmony with each other.

What distinguishes the second-order classes is that their members are first-order classes, but not the members of first order-classes. Thus, the lowest second-order classes ‘fan-out’ to first-order classes, but stop short of similarly relating to the members of first-order classes.

d. In conclusion, it is important to keep in mind that the concept of ‘inclusion’ has many meanings. It can mean inclusion of things in a first order class, or inclusion of first order classes in a second order class, or inclusion of a subclass in an overclass. These relations are not one and the same, though we call them all ‘inclusion’.

In practise, we are not always clear about the exact distinctions between subsumptives and nominals; first order classes (or simply, classes) and second order classes (or, classes of classes); equal-classes, subclasses and overclasses. But we have to be careful, because as we saw, their logical properties vary considerably.

3. Extreme Cases.

It is important to understand that the concepts of classes, or classes of classes, are purely relational. Although we colloquially use these expressions as if they were terms, there is no such thing as a ‘class’ which is not a class of certain things, or a class of classes of certain things. The word ‘of’ is operative here, and should not be ignored. It follows that we cannot say that classes are classes, or that “classes” is a class, or make similar statements, except very loosely speaking; we can only strictly say that such and such are classes of so and so.

Our habit of speaking of ‘classes’ or ‘classes of classes’ without awareness of the subtext, causes us to think that ‘classes’ is a collection of all classes, supposedly including all ‘classes of classes’ together with all ‘classes not of classes’, and even ‘classes’ itself and also ‘non-classes’. Similar ambiguity is generated by ‘classes of classes’. It is all very confusing, and due to the above mentioned imprecision.

If we want to think at once of the events of class-relating-to-its-members, we of course may do so. This is a class of all the ‘lines’ joining classes to their members (whether first-order classes to ultimate referents, or second-order classes to first-order classes). The resulting concept is, however, what we call ‘subsumption’ (or ‘membership’, in the reverse direction). If we want to think of hierarchical relations, we again may do so; but the resulting concept is again a copula.

If we want to speak of the terms of such relations, say, all classes indefinitely — that is, without having to specify what they are classes of — we strictly should say ‘the classes of anything’, where ‘anything’ is understood like a variable ‘X’, standing for any kind of thing we choose to substitute in its place. Likewise, for classes of classes (of anything), or with reference to hierarchies.

The largest possible class, is the class of all things (including real and illusory things), or simply “things”; it is not just ‘classes’. From our definition, since every thing is a thing, every thing is a member of “things” or the class of things. The largest possible class of classes, is the class of all classes of things, or simply the “things-class”; it is not just ‘classes of classes’. This means, again by definition, since all things are things, “things” (or the class of things) is a class of things, or a member of the “things-class”, or the class of classes of things.

Since a nominal (the class of anything) is itself a thing, it follows that the classes “things” and “things-classes” are both things, and so members of “things”. Additionally, since for any X, “X” is an X-class, it follows that “things” is a member of “things-classes”. Thus, exceptionally, the classes “things” and “things-classes” seem to be equal to each other and, somehow, members of themselves. They are (it is) the summum genus of all hierarchies.

When this summum genus branches out into species like “dogs”, “machines”, and such, it is preferably called “things”; when we focus on its subsumption, not of the ultimate referents, but of the ideational instruments standing between it and them, we call it “things-classes”; alternatively, we may embrace both these categories.