However, the tree of Figure 2.5 encourages a stronger objection. What distinguishes God from man is the difference mortal/immortal; but horse and man are both mortal and are distinguished by the difference ‘ration-al/irrational’. Therefore one has to choose either of the following alter-natives: (a) the differentia ‘mortal/immortal’ is not divisive of the genus ‘rational animal’ but, rather, of the genus ‘animal’ (but in this first case it would be impossible to tell the difference between man and God); (b) the differentia ‘mortal/immortal’ occurs twice, one time under ‘rational animal’ and one time under ‘irrational animal’.
GENERA AND SPECIES
Differentiae Differentiae
Substance
corporeal incorporeal
Body
animate
sensitive
rational
mortal
Living Being
Animal
Rational Animal
Man vs. God
inanimate
insensitive
irrational
immortal
FIGURE 2.5
Porphyry would not have discouraged this second decision, since he says that the differentia «is often observed in many species, as four-footed in many animals which differ in species» (18.20). This is a very important remark. Also, Aristotle says that, when two or more genera are subordinate to an upper genus (as it happens to man and horse insofar as they are both animals), nothing prevents them from having the same differentiae (Cat., ib.15ff; Topics, 4.164b. 10).
In Posterior Analytics (2.90bff) Aristotle shows how it is possible to ar-rive at an unambiguous definition of the number 3. Keeping in mind that for Greeks I was not a number (but rather the source and the mea-sure of any successive number), the number 3 can be defined as that odd number which is prime in both senses of being neither measurable by number nor composed of numbers. This definition is (biconditionally) equivalent with the expression three. But it is interesting to represent (Figure 2.6) how Aristotle reaches this conclusion by a careful work of division.
This sort of division displays two interesting consequences: (a) the properties registered in italic are not exclusive of a given node but can occur under more nodes; and (b) a given species (2, 3, 9) can be defined by the conjunction of more of the above properties. In fact, these prop-erties are differentiae. Thus Aristotle shows by a clear example, not only
numbers
even odd
sum and product of others
neither sum prime not prime nor product
of others
not sum of others
2
not product of others
2
not sum of others
3
not product of others
3 9
FIGURE 2.6
that many differentiae can be attributed to the same species, but also that the same couple of differentiae can occur under diverse genera. Moreover, he shows that, once a given difference has served to define unambigu-ously a given species, one is not obliged to take into account all the other subjects or which it is predicable: in other words, once one or more differentiae have served to define without ambiguities the number 3, it is irrelevant that they can also serve to define the number 2. (For a clear statement in this sense, see Post. An. 2.13.97315—25.)
But at this point we can try a further step. Once it has been said that, given subordinate genera, nothing prevents them from having the same differentiae, and since the tree of substances is wholly made of genera that are all subordinate to the uppermost one, it is hard to tell how many times the same couple of differentiae can occur. In Topics (1.15.106aff) Aristotle says that we can call both a sound and a material substance ‘sharp’. It is true that ‘sharp’ does not mean the same differentia in both cases, since in the former case it is opposed to ‘flat’ and in the latter case to ‘dull’ (thus the same name is used equivocally for two diverse differ-entiae). However, the same opposition ‘white/black’ occurs when one speaks either of colors or of sounds. Aristotle is convinced that this equivocality is a merely lexical one, since the same couple of contraries is referred to two different cases of sense-perception (sight and hearing). However, this equivocality is similar to the nonunivocal use of to be for the propositions men are animals (which is a matter of substantial being) and these men are white (which is a matter of accident): a case in which Aristotle speaks of «being used in various senses but with reference to one central idea» (πρός εν: Metaphysics 4.1003330). This reference to a central idea will be translated (by medieval philosophers) in terms of anal-ogy, of proportion, not in terms of mere equivocality.
In Topics 1.15.107b30) Aristotle will also say that the couple ‘white/
black’ when referred to a body is a species of color, whereas when referred to a sound it is a differentia (for one note differs from the other in being more or less clear or white). The whole matter is very complex, but how to avoid the suspicion that the entire universe of differentiae is polluted by metaphorical ambiguities (be they due to mere equivocality or to analogy)? Is ‘two-footed’ as referred to man the same as ‘two-footed’ as referred to a bird? Is ‘rational’ as applied to man the same as ‘rational’ as applied to God?
2.2.4. The tree is entirely made up with differentiae
Many medieval commentators on the Isagoge seem to encourage our suspicions. Boethius (In Isagogen, С. S. E. L, 256.10—12, 266.13—15) writes that ‘mortal’ can be a differentia for ‘irrational animal’ and that the species ‘horse’ is constituted by the differentia ‘irrational and mortal’. He also suggests that ‘immortal’ can be a differentia for celestial bodies which are inanimate and immortal: «In this case the differentia immortal is shared between species that differ not only in their proximate genus but also in all their genera up to the subaltern genus second from the top of the tree» (Stump 1978:257).
According to Stump, the suspicion aroused by Boethius is «surprising» and «disconcerting»; in fact, it is only reasonable. Aristotle and Porphyry say that a differentia is greater (encompasses more) than its subject, and this could not be if only men were mortal and only gods were immortals. In order to have such a result ‘mortal/immortal’ must occur under more than a genus, and so on for the other differentiae. Otherwise it would be incomprehensible why (according to Topics 4.144325) the species entails the difference, but the difference does not entail the species: ‘mortal’ has a wider extension than ‘animal rational and mortal’.
Abelard, in his Editio super Porphyrium (157V.15), suggests that a given differentia is predicated of more and different species: «falsum est quod omnis differentia sequens ponit superiores, quia ubi sunt permixtae dif-ferentiae, fallit.» Thus, given that (a) the same differentia can encom-pass more species, (b) the same couple of differentiae can occur under more than one genus, (c) different couples of differentiae can be repre-sented under many genera by using the same names (equivocally or analogically), and (d) it is an open question how high in the tree the common genus can be in respect to which many subordinate genera can host the same differentiae, then one is entitled to reformulate the Por-phyrian tree as in Figure 2.7.
The tree of Figure 2.7 displays many interesting characteristics: (a) it provides the representation of a possible world in which many new and still unknown natural kinds can be discovered and defined (for instance, incorporeal, animate, and irrational substances); (b) it shows that socalled
Substance
Corporeal
/body/
Incorporeal ?
Animate /living being/
Inanimate /mineral/?
Animate ?
Inanimate ?
Sensitive
/animal/
Insensitive ? ? ? ?
/vegetal/
Rational ?
Irrational
?
Rational
?
Irrational
?
Mortal Immortal
/man/ /God/
Mortal /brute/
Immortal ?
FIGURE 2.7
genera and species are names by which one labels only clusters of differentiae; (c) according to this tree, one cannot predict that if mortal then rational, or if irrational then corporeal, and so on; (d) as a conse-quence of (c) it can be freely reorganized according to alternative hierarchies.
As for the characteristic (a), we have seen what Boethius said apropos of celestial bodies. As for the characteristic (b), it must be clear that this tree is composed only of differentiae. Genera and species are only the names that we assign to the nodes represented by disjunctions of differentiae. Boethius, Abelard, and other medieval scholars knew this so well that they continually complain about the penuria nominum, that is, about the fact that we do not have enough lexical items to name the nodes of the tree represented by differentiae added to differentiae of upper differ-entiae.
Consider the case of the constitutive node obtained by adding the differentia ‘rational’ to the genus ‘animal’: the traditional tree labels this node with the expression rational animal, which is blatantly redundant and only repeats the name of the upper genus and of the differentia constitutive of the unnamed species. This laziness in providing names for species is rather inconceivable. The medieval imagination could have coined some new term. But it is a merely empirical circumstance that somebody found the name /animal/ for labeling the node composed by adding the difference ‘sensitive’ to /living being/, which is in turn the mere name of the composition ‘animate + corporeal’. The names of genera are insufficient because they are mere shorthand: a genus is no more than a cluster of differentiae. Genera and species are linguistic
ghosts that cover the real nature of the tree and of the universe it repre-sents: a world of pure differentiae.
Thus, if Aristotle did not list the species among the predicables (since the species is a sum of genus and differentia), for the same reason also genera could be eliminated, since they, too, are mere sums of upper genera and other differentiae.
As for the characteristic (c), since differentiae do not contain
