From the Tree to the Labyrinth (2 page)

The problem is coming up with the right attributes that can be predicated as elements of the definition (
Posterior Analytics,
II, xiii, 96a–b). Aristotle gives the example of the number 3: an attribute such as being certainly applies to the number 3, but also to anything else that is not a number. On the other hand, the fact of being odd applies to the number 3 in such a way that, even if it has a wider application (it also applies, for instance, to the number 5), it nonetheless does not extend beyond the class of numbers. These are the attributes we must look for “up to the point where, although singly they have a wider extension of meaning than the subject, collectively they have not; for this must be the essence of the thing” (II, xiii, 96a 35). What Aristotle means is that, if we define man as MORTAL, ANIMAL and RATIONAL, each of these attributes, taken on its own, can also be applied to other beings (horses, for example, are animal and mortal, and the gods, in the Neo-Platonic sense of the word, are animal and rational), but, taken altogether, as a defining “group,” MORTAL RATIONAL ANIMAL applies only to man, and in a way that is absolutely reciprocal.

A definition is not a demonstration: to reveal the essence of a thing is not the same as to prove a proposition about that thing; a definition says
what
something is, whereas a demonstration proves
that
something is (II, iii, 91a 1), and, consequently, in a definition we assume what a demonstration must on the contrary prove (II, 3, 91a 35). Those who define do not prove that something exists (II, iii, 92a 20). This means that for Aristotle a definition is concerned with
meaning
and has nothing to do with processes of
reference
to a state of the world (II, iii, 93b 30).

To find the right way to construct good definitions, Aristotle develops the theory of
predicables,
that is, of the ways in which categories can be predicated of a subject. In his
Topics
(I, iv, 101b 17–25) he identifies only four predicables (genus, proprium or unique property, definition, and accident), while Porphyry—as we shall see—will speak of five predicables (genus, species, difference, proprium, and accident).
¹

In a lengthy discussion in the
Posterior Analytics
(II, xiii), Aristotle outlines a series of rules to develop a proper division, proceeding from the most universal genera to the
infimae species,
identifying at each stage of the division the proper difference.

This is the method followed by Porphyry in the
Isagoge.
The fact that Porphyry develops a theory of division in a commentary on the
Categories
(where the problem of difference is hardly mentioned) is a serious matter for debate (see, for instance, Moody 1935), but it is not particularly relevant to our analysis.

In the same way, we may sidestep the
vexata quaestio
of the nature of universals, a question that Boethius bequeaths to the Middle Ages, taking the
Isagoge
itself as his point of departure. Porphyry declares his intention (we do not know how sincere he is) of setting aside the question of whether genera and species exist in and of themselves or if they are concepts of the mind. However that may be, he is the first to translate Aristotle in terms of a tree, and it is certainly difficult to avoid the suspicion that, in so doing, he is indebted to the Neo-Platonic notion of the Great Chain of Being.
²
We may safely ignore, however, the metaphysics that underlies the
Arbor Porphyriana,
given that what interests us is the fact that this tree, whatever its metaphysical roots, is conceived of as a representation of logical relationships.

Porphyry delineates a
single
tree of substances, whereas Aristotle uses the method of division with a great deal of caution and, we might add, a great deal of skepticism. He seems to give it considerable weight in the
Posterior Analytics,
but to be more circumspect in
On the Parts of Animals
(642b et seq.), where he gives the impression of being prepared to construct different trees depending on which problem he is dealing with, even when it comes to defining the same species (see the whole discourse on animals with horns, apropos of which see Eco 1983a).

But Porphyry outlined a single tree of substances, and it is through this model, and not the more problematical discussion in the real Aristotle, that the idea of a dictionary structure of definition is transmitted, via Boethius, down to our own day, even though present-day proponents of a dictionary-based semantics may not know to whom they are indebted.

Porphyry, we were saying, lists five predicables: genus, species, difference, proprium, and accident. The five predicables establish the mode of definition for each of the ten categories. It is possible, then, to imagine ten Porphyrian trees: one for substances, which allows us, for example, to define man as MORTAL RATIONAL ANIMAL, and one for each of the other nine categories—a tree of qualities, for example, in which purple is defined as a species of the genus red.
³
Therefore there are ten possible trees, but there is no tree of trees because Being is not a
summum genus.

There can be no doubt that the Porphyrian tree of substances aspires to be a hierarchical and finite whole of genera and species. The definition Porphyry gives of “genus” is purely formal: a genus is that to which a species is subordinate. Conversely, a species is what is subordinate to a genus. Genus and species are mutually definable and therefore complementary. Every genus placed on a high node of the tree includes the species that depend upon it; every species subordinate to a genus is a genus for the species subordinate to it, down to the base of the tree, where the
specie specialissime,
or “second substances,” such as man, for instance, are collocated. At the highest fork is the
genus generalissimum
(represented by the name of the category), which cannot be a species of anything else. A genus can be a
predicate
of its own species, whereas the species
belong
to a genus.

The relationship of species to their superior genera is a relationship of hyponyms to hyperonyms. This phenomenon would guarantee the finite structure of the tree since, granted a given number of
specie specialissime,
and given that for two (or more) species there is only one genus, then, as we proceed upward, in the end the tree inevitably tapers off till it reaches the root node. In this sense the tree would fulfill all the functions required of a good dictionary.

But a Porphyrian tree cannot be made up only of genera and species. If this were the case, it would take the form illustrated in
Figure 1.1
.

In a tree of this kind man and horse (or man and cat) could not be distinguished from one another. A man is different from a horse because, though both may be animals, the first is rational and the second isn’t. Rationality is the
difference
for man. Difference is the crucial element, because
accidents
are not required to produce a definition.
⁴

Differences may be separable from the subject (such as being hot, being in motion, being sick), in which case they are simply “accidents” (things that may happen—from the Latin
accidere
[
=
happen]—to a subject or not happen). But they may also be inseparable: among these some are inseparable but still accidental (like having a snub nose), others belong to the subject in and of itself, or essentially, like being
rational
or
mortal.
These are the
specific
differences and are added to the genus to form the definition of the species.

Figure 1.1

Differences may be divisive or constitutive. For example, the genus LIVING BEING is potentially divisible into the differences
sensitive/insensitive,
but the
sensitive
difference may be compounded with the genus LIVING to constitute the species ANIMAL. In its turn ANIMAL becomes a genus divisible into
rational/irrational,
but the
rational
difference is constitutive, with the genus that it divides, of the species RATIONAL ANIMAL. Differences, then, divide a genus (and the genus contains them as potential opposites) and they are selected to constitute in practice a subordinate species, destined to become in its turn a genus divisible into new differences.

The
Isagoge
suggests the idea of the tree only verbally, but medieval tradition visualized the project as seen in
Figure 1.2
.

In the tree in
Figure 1.2
the dotted lines mark the dividing differences, while the solid lines mark the constitutive differences. We remind the reader that the god appears both as an animal and as a body because, in the Platonic theology that constitutes Porphyry’s frame of reference, the gods are intermediary natural forces and not to be identified with the One.
⁵

Figure 1.2

From the contemporary point of view of a distinction between dictionary and encyclopedia, the Porphyrian tree certainly introduces, with its differences, encyclopedic properties into a dictionary structure. In fact, being
Sensitive, Animate, Rational,
and
Mortal
are accidents identifiable in terms of knowledge of the world, and it is on the basis of its behavior that we decide whether a being is animate or rational, whether, in other words, it expresses ratiocinative capabilities by means of language. In any case, the end purposes of the tree are those of a dictionary, in which the differences are necessary and sufficient conditions to distinguish one being from another and to make the
definiens
or definer coextensive with the
definiendum
or definee
,
so that, if ANIMAL RATIONAL MORTAL, therefore of necessity
human,
and vice versa.

Once more, however, in its canonical version, this tree reveals its inadequacy, because it distinguishes, in a logically satisfactory fashion, God from man, but not, let’s say, a man from a horse. If we had to define the horse, the tree would have to be enriched with further disjunctions: we would need, for example, to divide ANIMALS into
mortal
and
immortal,
and the next species down—that of MORTAL ANIMALS—into
rational
(men) and
irrational
(horses, for instance), even though, unfortunately, this subdivision, as is apparent in
Figure 1.3
, would not allow us to distinguish horses from donkeys, cats, or dogs.

Figure 1.3

Even if we were willing to pay this price, however, we still could not reintroduce God into the tree. The only solution would be to insert the same difference twice (at least) under two different genera (
Figure 1.4
).