Resumen: El artículo discute el papel tanto de las categorías supremas como el de las subcategorías a partir de la tradicional distinción aristotélica entre una división mínima y una división máxima del ente. Señala que para determinar las categorías son necesarias las propiedades categories. Se argumenta que un existente no puede poseer dos propiedades esenciales de tipo categorial, mientras que solo las subcategorias más bajas en la escala poseen propiedades categories simples. Se señala, además, que las categorias y las subcategorias se organizan en un árbol según una cierta unidad y que, por el contrario, los conceptos no se jerarquizan necesariamente formando un árbol. También se analizan las dificultades que Porfirio y Simplicio encontraron en la división mínima y máxima de Aristóteles. Finalmente, el artículo critica el modo en que Aristóteles evita, a través de la abstracción, la referencia a las propiedades categories.
Palabras clave: Categoría, subcategoría, Árbol de Profirió, división minima y maxima, propiedad categorial.
Abstract: Starting from the traditional distinction between the minimal and the maximal division, the role of subcategories in Aristotle, as well as that of the highest categories, is discussed. The need for categorial properties which determine categories is pointed out. It is argued that an existent cannot have two such essential properties and that only the lowest subcategories have simple categorial properties. Furthermore, it is emphasised that categories and subcategories must form atree because they belong to a theory of categories which requires unity. By contrast, it is held that the hierarchy of all concepts need not form a tree. The difficulties Porphyrius and Simplicius find in Aristotle's minimal and maximal division are analysed. Finally, Aristotle'sway of avoiding categorial properties by referring to an abstraction is criticised.
Keywords: Category, subcategory, Porphyrian Tree, Minimal and Maximal Division, categorial property.
Categories and Subcategories
In Aristotle's book Categoriae there is a minimal division of all existents into two categories, namely substances and accidents and a maximal division into ten categories, namely primary substances, secondary substances, quantities, relations, qualities, places, times, states, actions, and affections. Why this two-fold division and how do the two divisions go together? The latter are subcategories of the former. Thus the question is again why categories as well as subcategories are needed. Wouldn't the subcategories suffice?
Since categories are classes and since, obviously, they are not merely enumerative classes but rather extensions there arises the need for categorial properties of which the categories are extensions. Moreover, a clash between the categorial properties threatens because existents are necessarily members of the upper categories as well as of their subcategories. Categorial properties should be necessary properties and such properties have to be very closely connected. That poses the question whether an existent can have more than one categorial property and if not whether the maximal and the minimal division are compatible or how they can be reconciled. Thus, the question is also whether it is even consistent to advocate both a minimal and a maximal division.
The minimal division has a problematic entailment if it is taken to be the highest level of the categorial hierarchy. It entails that there is no top and thus no categorial tree but rather a categorial forest with several trees. The maximal division is the lowest categorial division and thus the borderline between categorial and noncategorial division. Therefore the question arises why to draw the borderline there and not somewhere else.
I should make clear that the paper does not offer a detailed historical representation but rather a rational and systematic reconstruction of the project of a theory of categories as a hierarchy of classifications. The most influential conception of this project originated from Aristotle. The following discussion revolves around the constraints of classificational hierarchies (also called "Porphyrian Trees").
Consider a very simple class hierarchy. The class of triangles is subdivided into equilateral and scalene triangles. Thus the classes of equilateral and scalene triangles are subclasses of the class of triangles. One can represent that in a tree graph which visualises the talk of a hierarchy. The branches of the tree (mathematicians say "edges" (1)) are formed by the subclass relation. Now, the designations of the classes refer to certain properties which its members have. I hold that classes normally are determined by properties. Which are the properties determining the three classes of our example? The upper class of the hierarchy seems is determined by the property of having three angles. That is what the designation "triangle" implies. The lower classes are presumably determined by the conjunctive property of having three angles and of having sides of the same length and by the conjunctive property of having three angles and of having sides of differing length. Obviously, the lower properties contain the higher properties as conjunctive parts. Thus the branches of the tree graph can also be interpreted as part relations between properties and the vertices of the tree would then be the properties determining the classes. Furthermore, all the lower properties would be conjunctive properties. Only the highest property could be a simple one. Thus we would, strictly speaking, have two tree graphs which are congruent: a tree graph of classes and a tree graph of properties. They would be rather small trees and they would not be trees of categories. But they can possibly be integrated into a huge tree encompassing all kinds. The top area of that tree would be occupied by categories. The size of the top area is variable depending on the respective ontology. It might be only the highest vertex and a two-fold division below or it might comprise more levels of vertices and longer chains of edges. The huge and all-encompassing tree has traditionally been called the "Porphyrian Tree".
Like in any tree graph there are no circles in a Porphyrian tree, i.e., if one moves down one will not return to any vertex. Although branches (edges) run only vertically the graph not only shows subordination and superordination but also co-ordination of vertices. Vertices on the same level and subordinated to the same next upper vertex are said to be co-ordinated. Co-ordinated vertices form classifications together and thus it holds that the respective classes are disjunct (i.e. they don't have any members in common) while their union class is identical with the class of the next superordinate vertex. As a consequence it holds that co-ordinate categories always form a classification together. Thus co-ordinate categories exclude each other, i.e., they imply the negation of each other. Substance and accident in Aristotelianism, for example, are co-ordinate categories and it is assumed that a substance cannot be an accident and an accident cannot be a substance. If they form a categorial classification together, then it follows also that any existent must be either a substance or an accident.
CATEGORIES AND DEFINITIONS
The above paragraph contains the view that categories are classes and that they are determined by properties. The view is not exactly Aristotle's but it is a clear view and it allows explicating and discussing Aristotle's view. The properties which determine the categories will because of their role be called "categorial properties". A subcategory is subclass of the category of which it is subcategory. Looking at a Porphyrian Tree one realises...