Aristotle's Principle of Non-Contradiction

Hi Aristotle Friends,

I am investigating the impact of the Principle of Non-Contradiction (PNC) for a first science or science of science. The PNC is for Aristolte according to the Metaphysics the highest principle. For this reason I would like to know how the PNC can be demonstrated in its importance on a variety of subjects.

Right now I am intressted in the importance of the PNC for the generation of content. According to contemporary theorists PNC is not merely a formal condition, but also a condition for the content at all. I found the following passage in Brandom that makes a mysterious reference to Aristotle. I would like to know your thoughts on this, and how it might relate to Aristotle:

"So within the scope of his commitment to understanding determinateness of content in terms of material incompatibility, Hegel sees a structured relationship between the notion of incompatibility and that of objecthood. We may, as Hegel does, talk about what is materially incompatible with a property as an ‘opposite’ of it, provided we remember that in this sense things can have more than one ‘opposite’ (since white is thus opposed not only to green, but also to blue, and yellow, and so on). Then the basic structural observations being made about objects and properties take the form of deep relations between Gegensätze (opposites) and Gegenstände (objects). Properties are defined by having opposites, in this sense, and objects are the opposite of what has opposites. For the only sense in which objects ‘exclude’ one another is that two objects can have properties that are incompatible with each other. But this is what mere difference, weak contrast, among objects consists in. And for objects there is no further sense of strong contrast. For as we have seen, wherever material incompatibility is definable, so is formal contradiction. The contradictory of a property (or claim) is its least incompatible, what is entailed by everything incompatible with that property. The contradictory of a property P will then be a property exhibited by all and only those objects that do not exhibit P. The contradictory of an object O would be an object that exhibited all and only the properties not exhibited by O. ********But, as Aristotle already realized, this notion makes no sense.******* For many of the properties (not just different from but) incompatible with properties of O will be (not just different from but) incompatible with each other. If O is white, its contradictory would have to be green and blue and yellow, and so on. If O is not identical to the number 17 and not identical to my left big toe, then O would have to be both identical to the number 17 and identical to my left big toe. And so on. "

Again, how might this relate to Aristotle? I am unclear on that.