Previously in the Categories, Aristotle has shown that the name and definition of the primary substance are necessarily predicated of that substance. Now he shows that all other subjects are predicated of primary substances.
All the other things are either said of the primary substances as subjects or in them as subjects. This is clear from an examination of cases. For example, animal is predicated of man and therefore also of the individual man; for were it predicated of none of the individual men it would not be predicated of man at all. Again, colour is in body and therefore also in an individual body; for were it not in some individual body it would not be in body at all. Thus all the other things are either said of the primary substances as subjects or in them as subjects. So if the primary substances did not exist it would be impossible for any of the other things to exist.
All names that do not refer to primary substances are either said of primary substances or are in primary substances. This can be proven by examining a wide variety of examples. For example, animal is predicated of man when we say “men are animals”. But individual men are animals too. If no individual men were animals then men could not be animals at all. When we claim that middle sized objects are colored, then color is in some middle sized object too. If color were in some middle sized particular thing, then it would not be in middle sized objects at all either. Therefore, all subjects are either said of primary substances or are in primary substances. If primary substances did not exist, then nothing else would exist either.
For Aristotle, the primary and ultimate things that exist are particular things. Everything else is either a part of them or exists in them. There are no abstract objects that exist in some Platonic heaven.
Next, Aristotle explains why secondary substances are substances.