RUSSEL’S PARADOX AND GREEK ONTOLOGY
(Rodrigo Peñaloza, March 2015)

Mathematics is intimately related to Ontology. It is a mistake to separate mathematical reasoning from philosophical thinking in all instances. One way to see this is by means of Russel’s Paradox. I will explain first what this paradox is, then I will argue how the ancient Greeks got to it through Ontology, not through Mathematics, and how it is related to the idea of being, that is, τò ὄν.

Let X be the set of all sets that are not members of themselves. What can we say about X? There are two alternatives: either X is a member of itself or it is not. If X is a member of itself (of X), then (by the very definition of X) it is not a member of X. On the other hand, if X is not a member of itself (of X), then it is a member of X. In other words: if X is a member of X, then it is not a member of X; if X is not a member of X, then it is a member of X. In any case, we have a contradiction! This is Russel’s paradox.

What does Ontology have to do with it? Consider the set X of all things that are not, that is, the non-being. Is X a being? This question is relevant, because we are talking about it and we want to know if it makes sense to talk about it. Can we say that “X is”? Suppose X is not a member of itself (that is, that the non-being “is not” a non-being). Then it is not a member of X, hence X is. In plain words, if X is not, then it is. Suppose, on the other hand, that X is a member of itself (of X), that is, that non-being is not non-being. Then it is being, so it is something. In any case, we have a contradiction: the being is not; the non-being is.

What is the problem? In philosophical terms, in our search for “being”, when we consider a species as a subset of a genus, we tend to consider this genus as a subset of a higher genus, and so on. The naïve thinker will assume that being is the highest genus, in relation to which all genera will be mere species. This is false!

Mathematicians (Zermelo and Russel) solved Russel’s Paradox by distinguishing between sets and classes. A class X is a collection of sets satisfying a property P, with the proviso that P be so given in such a way that any set either does satisfy it or does not. With this proviso, we exclude sets that simultaneously satisfy P and not-P. The lesson is, we cannot generalize by taking sets of sets indefinitely. We have to stop at the idea of classes.

In Ontology, this means that we cannot consider being as a generalization. It doesn’t make sense to follow the path “Socrates → man → mammal → animal → being” neither the path “Kiki → parrot → bird → animal → being”, as if “being” were the highest “generalization”. The “being” is not a genus. In order to go from a generic notion (defined in terms of some property P) down to a particular notion, that is, from a genus down to one of its species, we have to add new properties that are not part of the initial property. This was called by Aristotle specific difference. For example, in order to go from the genus “animal” down to its species “man”, we have to add the property “rational”. Being is not a genus because we cannot add any specific difference that is not already part of the idea of being. Note that by requiring the idea of specific difference in order to go down from a genus to one of its species, what Aristotle is doing is exactly what Zermelo and Russel did in order to solve Russel’s paradox. It is property P that specifies the specific difference that allows for the specification of a genus.

It was Aristotle who went deep into these concepts to clear up the debate. Before him, when Parmenides said, “the being is, the non-being is not”, he wasn’t saying trivialities! He was alerting us to stop searching the being by means of generalizations. He was thinking of classes.

Once again, Greece 1 x 0 World.

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