The structure of definitions in Aristotle, according to Christian Pfeiffer (and me)

Complexity is both a sufficient and necessary condition of definability: every complex is definable and only complex entities can be defined. But what is a complex entity? Obviously, an entity is complex if it is not simple, i.e., it must contain more than one component.This is what Pfeiffer calls the complexity constraint. Complexity is a necessary condition, but not a sufficient one, for complex entities are not mere pluralities (as, for example, heaps of objects – an example that both Pfeiffer and me favor): they have unity. In other words, they are not just complex, but complex entities. Thus, complex entities are things that satisfy both the complexity and unity constraints.

Now, how do we define these complex things?

Assume the simplest of complex entities: an ordered pair AB. It is complex, thus it must be definable. According to Aristotle (according to Pfeiffer) is to say that it is the sequence of A and B in that order. Here, it is very easy to identify the matter (what are the defined entity’s constituents) and the form (how they are arranged to compose the defined entity). The matter is A and B, and the form is the order of A being first and B being second. According to Aristotle, every definition has this same, basic structure.

This is why, in strict sense, neither matter (DA III.6 430b27-9; Metaph Z.4 1030a6-11, etc.) nor form can be defined. In order to understand why it is important to keep in mind Aristotle’s point that we sometimes use the same term to talk about the complex entity (what I call a “system” by the way) and its form (H.3 1043a29-37): Because matter by itself could be composed into many different things, but once matter takes a certain form, it cannot be but a single sort of thing. Form, in contrast, even when it can inform different matter, the result is always the same sort of object. Thus, what makes any pair a pair is the same (they share the same form of pairness) regardless of its components (regardless of their matter, that is). This asymmetry between form and matter is fundamental and that is why even though neither forms nor matter can be defined (only complex entities can) we only get confused in thinking that forms can be defined and not matter.

Some people have made this point in terms of universality: forms are universals and matter is purely particular, but I am not sure that is a good way of making this very important point.

The form is predicated of the matter. (H.3 1043 b23-32) It is very clear that the relation between form and matter cannot be one of combination or composition (H.3 1043b4-14) for that would entail Bradley’s famous infinite regress (Z.17 1041b16-22). The combination of form and matter would then be the matter of a new complex that would require a new form to be defined and so forth ad infinitum.

Earlier, I said that the matter of the ordered pair A and B was A and B. But this seems to mean that the matter of the pair is already complex; after all, it is already a pair. Wouldn't that require a further definition, since it is also a complex entity – it satisfies the complexity requirement (since it has two components A and B) and the unity constraints (since it is a pair, different from others; it has identity criteria that allows us to distinguish this pair from others, like B and C, or D and E)? If Aristotle were to answer in the positive, he would fall into another vicious regress. Thus, he is forced to insist that the pair is undefinable and that it is not a complex entity. This means that he has to reject either its complexity or its unity, i.e., it has to argue either that it is not actually complex, it does not satisfy the complexity constraint, or that it lacks actual unity, it does not satisfy the unity constraint. Aristotle goes for the second one, even though some other philosophers have also rejected the first one arguing that before being ordered, they are not a pair. Thus, for Aristotle, unordered pairs, unlike ordered ones, are not actual complex entities because even if they are not simple, they are not complex either.

But, wait, what is then this unordered pair A and B if it is neither a simple nor a complex? In other words, Aristotle gives the bite the bullet answer that, well, it is nothing, i.e., it is not a substance, but it is still real (According to Pfeiffer, the textual evidence is ambiguous at this point).

Pfeiffer likes making this point in hierarchical terms. Pluralities, heaps, lists, do not have real unity, they have quai-unity. I think this raises just as much issues as it solves, so I am weary of this sort of answers (as I have written elsewhere).

But the problem raises as well for forms themselves. Again, looking at order pairs, it is clear that the form of an ordered pair itself has parts: it contains A being the first and B being the second. As a matter of fact, Aristotle goes as far as to hold that definitions must be isomorphic to the matter they inform in so far, thus they must be complex themselves.