Aristotle has just shown that what is predicated of secondary substances and differences is predicated of the primary substance synonymously. The Categories continues by showing that substance is a unity.
very substance seems to signify a certain ‘this’. As regards the primary substances, it is indisputably true that each of them signifies a certain ‘this’; for the thing revealed is individual and numerically one. But as regards the secondary substances, though it appears from the form of the name—when one speaks of man or animal—that a secondary substance likewise signifies a certain ‘this’, this is not really true; rather, it signifies a certain qualification—for the subject is not, as the primary substance is, one, but man and animal are said of many things. However, it does not signify simply a certain qualification, as white does. White signifies nothing but a qualification, whereas the species and the genus mark off the qualification of substance—they signify substance of a certain qualification. (One draws a wider boundary with the genus than with the species, for in speaking of animal one takes in more than in speaking of man.)
It seems that all substances point out a certain kind of thing that can be spoken of as a unity. This is most certain with primary substances because they each one of them is an individual. Secondary substances are different because there are more than one of them. Two things can be men, even though each one of them is different than the other. So secondary substances are a way of speaking about a primary substance. But white is also a way of speaking about a primary substance. The difference between being human and being white is that being white simply adds something to the thing while being human is a way that the thing is in itself.
This is a fairly difficult passage to interpret. The line I have taken in doing so is that ‘this’ refers to a kind of unity. So secondary substances modify the thing as a whole. Whiteness, along with other characteristics in the subject, do not modify the thing as a whole. They are added on to the thing so that they can be separated from it by analysis. Secondary substances cannot be separated because the primary substance is truly an instance of that secondary substance. It cannot be separated from the secondary substance in any respect. The secondary substance and it form a kind of unity. Secondary substances do not have the kind of unity that primary substances have because there are many secondary substances that are all equal kinds of that substance, but only ever one primary substance that is identical. Given this interpretation, Aristotle does not offer proof. I understand his goal in this case to be explaining certain features of our speech. At this point, Aristotle does not set out to prove that secondary substances are real, but only to explain how they are different from what is “in” substances. I think that it is best to understand Aristotle as setting out the implications of a certain understanding of the world, given our speech about the world. That our speech is generally reliable and accurately divides the world is something that Aristotle has assumed along with his contemporaries.
Next, Aristotle shows that substance has no contraries.