On Ayn Rand's failure to examine "class-formation"

References to ITOE2 refer to Ayn Rand's "Introduction
to Objectivist Epistemology," second edition.

Let me start off by defining what I mean by a "class."

A class is "a collection of things sharing a common attribute."
http://wordnet.princeton.edu/perl/webwn

An example of this would be the class of red books. Notice that I have
not defined a concept here, only a class. It's distinguishing
characteristic is redness, which is a universal, but not essential to
forming a special concept of red books. (There is also the universal
'bookness' characterizing the books, but let's just discuss redness
for now.)

In order to designate a concept it was necessary for Rand to designate
a special form of class, known as a concept, which integrates two or
more instances by means of certain properties known as essentials.
These essentials are defined as distinguishing characteristics of an
entity, distinguished "from the other entities belonging to the same
genus." (ITOE2, 41)

(I'll ignore the fact that her theory of concept-formation by
essentials illustrates a logical fallacy known as the 'fallacy of
difference,' which states that distinguishing characteristics are not
necessarily essentials, as rationality is not necessarily essential to
manness merely because it is a characteristic used to distinguish
man from animals. Or perhaps let's just say rationality is only
essential to forming the concept, it is not necessarily essential in
the metaphysical sense.)

In class theory it is not necessary to denote two or more instances in
forming the class. A class can consist of one member or even zero, for
example, if someone painted an elephant pink, and it was the only one
in the world, the only member of its class; or if there were no pink
elephants in existence, the class would be empty.

No attribute is essential to forming a class, what matters is that the
uniting characteristic is a universal, that is, the attribute is shared
by all members of the class such that it helps to define the class.
In that case, the attribute is a property, belonging to individual
entities as members of the class, becomes known as a universal.
A universal is held in common by all members of a class, even if the
class consists of only one member, or none.

ITOE consists of some kind of answer to the problem of universals,
accomplished, in Rand's own words, "by presenting my theory of
concepts" (Ibid, 3). And in her mind, universals are equated by Rand
with concepts or abstractions (1), reducing the metaphysical
problem to a problem of concepts.

So as a result of Rand neglecting to examine the issue of classes upon
which is genetically dependent the issue of concepts, she did not
thoroughly examine the issue of universals connected with it, assuming
the problem of universals is only an epistemological one (which it is
not; it is primarily metaphysical at its root). And if a particular
property can be universal to a class consisting of only one member,
then it follows that a concept can be formed from a class consisting
of only one member.

There is an example of such a class found in ITOE2, and that is the
concept or axiom "existence." It seems to violate Rand's rules of
concept-formation, particularly that of the conceptual common
denominator, "The characteristic(s) reducible to a unit of
measurement, by means of which man differentiates two or more
existents from other existents possessing it." (14) In this case there
is only one existent, existence itself. (For those of you who believe
that "existence" is only a concept, Rand believed that concepts are
also existents, indeed, even concrete mental entities.[156])

At one time I was led to believe that the axiomatic concept 'existence'
was no concept at all, that is, by Rand's own rules of concept-formation
(which she rationalized as a "special" exception [58-59]). However,
I now realize that the problem lies with her theory of concept-formation,
most basically, her neglect in examining the wider genus determining the
very notion of a concept, and that is, the class theory behind it.

Rand defined "concept" in one place as "a mental integration of
two or more units which are isolated according to a specific
characteristic(s) and united by a specific definition."(9) However,
"a mental integration" is not specific enough, as "set" or "class"
also constitute mental integrations of material serving as members
distinguished by properties held in common, that is, universals.
("Class" and "set" are often used interchangeably, but "set" is more
of a mathematical expression to which I don't want this analysis to
be limited.)

How does this issue apply to the problem of universals? This problem
does not rely on the issue of class or even concept, but how man has
knowledge of 'redness' despite the fact that, in reality, there is no
particular color that he can distinguish as epitomizing "redness." Or
perhaps many things can serve this function; however, all of them will
vary in their particular shade of red. Thus the question is not "where
is the manness in men" (2), but "How does man's knowledge of manness
arise?" Or, "How does man's knowledge of universals arise from amidst
the great variety of particulars around him?" This is a question of
epistemology, and it is not answered by any theory of
concept-formation.

Geometry can help answer this question, or at least, pose the question
in a more useful way. For example, there are no straight lines in
reality, none at any rate which correspond to man's concept of a
straight line. A straight line is a mental construct, an ideal
concept, used for creating or defining approximations of straight
lines in reality based in utility. Every straight line in reality
instantiates the universal "straightness," without however
substantiating it. There is no substantiating a universal because
there is no substance, no entity in reality equal to this mental
entity, only approximations to it. There is, to put it in terms of
Rand's question (2), straightness in a straight entity, but only
insofar as a mind puts it there. The universal therefore lies a priori
to the judgment of that which is considered straight.

And so, as Aristotle indicated in his Metaphysics II §8, Being and
Unity, the two highest universals, are as one in the same entity, only
I say not in the ontological sense. The being, the whatness, the
universal, is a product of the mind, the unity is universals to any
particular individual in reality (a unit). The product of their
synthesis is variety in experience, without which, in Aristotle's
words, "there is much difficulty in seeing how there will be anything
else besides these, - I mean, how things will be more than one in
number." For if Being and Unity, two universals, existed in
themselves, as properties of an entity, then it would be impossible
to have the concepts of plurality, variety, or many-ness.

And so the properties we designate to individuals, when these
individuals are formed into classes, become universals, but are not
thereby granted existence outside a thinking being qua universals.
Induction only gives the material for creating the universal, but as
essential to its completion there must be concepts borrowed from
metaphysics, that is, metaphysical properties attributed to the
members of the class. For example, Being (Substance), Unity, and
Necessity. Metaphysical concepts grant objective truth to inductive
reasoning, and not merely relative truth.

This class of concepts was not derived from empirical reality, as if
we could somehow intuit its Platonic form, they are nevertheless
metaphysical as deriving from man's primary *form* of cognition which
is rational. Just as man's metaphysical nature demands that he find
straightness in that which is not, as a formal standard to serve
practical building purposes, so his metaphysical form of reasoning
demands that he produce concepts of things from the vastness and
variety of nature, utilizing concepts of his own devising a priori.