HIERARCHIES AND ORDERS.
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).
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.
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.
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.
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.
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.
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
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.