Something that gets translated as “relation” (ta pros ti, literally “the toward something”) is one of the ten categories Aristotle discusses in the Categories, which was traditionally treated as a kind of introduction to Aristotelian logic, and indeed to Aristotle’s thought as a whole.
In the order of the sciences laid out by al-Farabi, for instance, I believe the Categories is treated as a source of primitive definitions along the lines of the definitions with which the systematic development of Euclid’s Elements of Geometry begins. This is to substitute a very different — straightforwardly deductive — method and pedagogy for Aristotle’s own more fluid approach. See Demonstrative “Science”?.
Plato and Aristotle devoted extraordinary attention to questions of definition, and in doing so greatly devalued the importance of any assumed definitions.
Aristotle always recommended that we begin with that is more familiar and close at hand, and then expect our beginning to be substantially modified as we move toward what is clearer and more intelligible. This is the original model for Hegel’s logical “movement”.
The “what is toward something” of the Categories is quite simply not equivalent to more modern notions of “relation” — neither to its use in Kant and Hegel, nor to its mathematical use. Whether in Kant or Hegel or in mathematics, relation in the modern sense is fundamentally bi-directional. If a has a relation R to b, then b by definition has a relation R-inverse to a. In the same sense in which Hegel points out that the positive and negative signs on numbers assigned to measure, e.g., physical forces, can be systematically reversed without changing the physical meaning, any directionality in relations in the modern sense is a superficial matter of setup, and not anything deeply meaningful.
On the other hand, Aristotle’s “what is toward something” has an irreducibly directed (i.e., unidirectional) character. If x is oriented “toward” y, it does not follow that y must have a corresponding inverse orientation toward x. The semantics of x‘s “being toward” y imply a material dependency of x on y, and thus implicitly a kind of subordination of x to y.
This is certainly an important kind of construct to have in our toolbox for explaining things, but it simply is not what is meant when Kant says we know phenomena in a purely relational way, or when Hegel adds that essence is purely relational. It would also be a serious error to assume that according to Aristotle, the subordinate or subordinating aspect of the pros ti category would apply to the different concept of “relation” used by Kant and Hegel (or to mathematical relations).
Once again, this whole confusion arises due to the influence of the Latin translation, in this case of pros ti by relatio. For Latin readers, relatio had not yet acquired the importantly different meanings that “relation” has in Kant and Hegel, or in the mathematical theory of relations pioneered by C. S. Pierce and Ernst Schröder. Thus its use did not create serious misunderstanding. But for a general modern audience, “relation” is a terrible choice to translate pros ti, for the reasons mentioned.
I think that Aristotle does also implicitly operate with a concept like that of “relation” in Kant and Hegel, but he does not give it a name, and it is certainly not the pros ti of the Categories. Rather, it comes into play in the way Aristotle uses notions like unity, diversity, identity, and difference.
The Postmodern Peripatetic 