From the Tree to the Labyrinth (3 page)

Figure 1.4

Porphyry would not have discouraged this decision, given that he himself says (18.20) that the same difference “can often be observed in different species, such as having four legs in many animals that belong to different species.”
⁶

Aristotle too said that when two or more genera are subordinate to a superior genus (as occurs in the case of the man and the horse, insofar as they are both animals), there is nothing to prevent them having the same differences (
Categories
1b 15 et seq.;
Topics
VI, 164b 10). In the
Posterior Analytics
(II, 90b et seq.), Aristotle demonstrates how one can arrive at an unambiguous definition of the number 3. Given that the number 1 was not a number for the Greeks (but the source and measure of all the other numbers), 3 could be defined as that odd number that is prime in both senses (that is, neither the sum nor the product of other numbers). This definition is fully reciprocable with the expression
three.
But it is interesting to reconstruct in
Figure 1.5
the process of division by which Aristotle arrives at this definition.

Figure 1.5

This type of division shows how properties like
not the sum
and
not the product
(which are differences) are not exclusive to any one disjuncture but can occur under several nodes. The same pair of dividing differences, then, can occur under several genera. Not only that, but the moment a certain difference has proved useful in defining a certain species unambiguously, it is no longer important to consider all the other subjects of which it is equally predicable (which amounts to saying that, once one or more differences have served to define the number 3, it is irrelevant that it may occur in the definition of other numbers).
⁷
Once we have said, then, that, given several subordinate genera, nothing prevents them having the same differences, it is difficult to say how many times the same pair of differences can occur.

In his
Topics
too (VI, 6, 144b), Aristotle admitted that the same difference may occur twice under two different genera (as long as they are not subordinate): “the earthbound animal and the flying animal are in fact genera not contained the one within the other, even though the notion of two-leggedness is the difference of both.”
⁸

If the same difference can recur a number of times, the finiteness and logical purity of the tree—which runs the risk of exploding into a dust cloud of differences, reproduced identically under different genera—are compromised. Indeed, if we reflect that species are a combination of genus and difference, and the genus higher up is in its turn a combination of another genus plus a difference (and therefore genera and species are abstractions, intellectual figments which serve to sum up various organizations of differences or accidents),
the most logical solution would be for the tree to be made up solely of differences,
properties that can be arranged into different trees according to the things to be defined, jettisoning the distinction between substances and accidents.

Many medieval commentators of the
Isagoge
appear to endorse this conclusion. Boethius in his
De divisione
(VI, 7) suggests that substances like pearl, ebony, milk, and some accidents like white or liquid may give rise to alternative trees. In one, for example, given a genus Liquids, with the differences
White/Black,
we would have the two species Milk and Ink; in the other, the genus White Things, with the differences
Liquid/Solid,
would generate the two species of Milk and Pearl (
Figure 1.6
).

True, in this passage Boethius is speaking only of accidents, but, in
De divisione
XII, 37, he applies the same principle to all divisions of genus (“generis unius fit multiplex divisio” [“a single genus is divisible in more than one way”]).
⁹

Figure 1.6

Abelard says the same thing in his
Editio super Porphyrium
(150, 12), where he reminds us that “Pluraliter ideo dicit genera, quia animal dividitur per rationale animal et irrationale; et rationale per mortale et immortale dividitur; et mortale per rationale et irrationale dividitur” (“He [Porphyry] refers then to genera in more than one way, for animal is divisible into rational animal and irrational animal; and rational is divisible into mortal and immortal; and mortal is divisible into rational and irrational”) (
Figure 1.7
).
¹⁰

Figure 1.7

In a tree composed solely of differences, these can be continually reorganized following the description under which a given subject is considered, and the tree thus becomes a structure sensitive to contexts, not an absolute dictionary.

On the other hand, when Aristotle (who is interested in defining accidents as well as substances) asserts (
Posterior Analytics
I, 3, 83a, 15) that definitions must stick to a finite number of determinations, in either an ascending or a descending series, he does not in the least seem to be suggesting
that their number and function are already established by a previous categorical structure.
In fact in his various researches into natural phenomena, from the eclipse to the definition of ruminants, he shows a great deal of flexibility in setting up subdivisions and suggesting trees in which genera, species, and differences exchange roles according to the problem one intends to resolve.

In
Posterior Analytics
II, 3, 90a, 15, he says that the eclipse is a deprivation of the sun’s light by the earth’s interposition. In order to define it this way we must suppose a division into genus and species like the one in
Figure 1.8
.

Figure 1.8

But what is the deprivation of the sun’s light a species of? Are we talking about a tree that takes cognizance of the various kinds of deprivation (among which, let’s say, are the deprivation of food and of life) or a tree that takes cognizance of various astronomical phenomena and opposes the radiation of the sun’s light to its deprivation?

In II, 3, 93b, 5, the example of thunder is discussed. It is defined as extinction of fire in the clouds. Hence a tree as in
Figure 1.9
:

Figure 1.9

But what if the definition had been “noise produced by the extinction of fire in the clouds”? In that case, the tree would have to look like
Figure 1.10
.

As can be seen, in the first case thunder is a species of the genus extinction, in the second case of the genus noises.

Figure 1.10

This flexibility is due to the fact that, when he is dealing with concrete phenomena, it is the philosopher’s intention
to define them,
while a tree with a fixed hierarchy and a finite number of determinations serves only to
classify.
Merely classificatory, for example, is a device that embeds genera, species, and differences without explaining the nature of the
definiendum.
This model is that of the taxonomy of today’s natural sciences, in which it is established, for instance, that a dog belongs to the genus CANIS, of the family of CANINES, of the suborder of FISSIPEDS, of the order of CARNIVORES, of the subclass of PLACENTALS, of the class of MAMMALS. This classification, however, does not tell us (and is not meant to tell us) either what the properties of a dog are or how to recognize a dog or refer to a dog. Every node of the classification is in fact a
pointer
that refers to another chapter of zoology in which the properties of Mammals, Placentalia, Carnivores, Fissipeds, and so on are specified.

A dictionary classification, then, does not serve to
define
a term but merely to allow us to use it in a logically correct fashion. Given, let’s say, that the imaginary order of the Prixides is classified as belonging to the genus Prosides and the Prosides are a species of the genus Proceides, we do not need to know what the properties of a proceid or a prosid are to draw (true) inferences along the lines of:
if this is a prixid then it has to be a prosid,
and
it is impossible that something that is a prixid should not be a proceid.