Aristotle has just shown that secondary substances and differences are a part of a primary substance in a different way than other things are in primary substances. Now he continues the Categories by showing that secondary substances and differences are always said synonymously.
It is a characteristic of substances and differentiae that all things called from them are so called synonymously. For all the predicates from them are predicated either of the individuals or of the species. (For from a primary substance there is no predicate, since it is said of no subject; and as for secondary substances, the species is predicated of the individual, the genus both of the species and of the individual. Similarly, differentiae too are predicated both of the species and of the individuals.) And the primary substances admit the definition of the species and of the genera, and the species admits that of the genus; for everything said of what is predicated will be said of the subject also. Similarly, both the species and the individuals admit the definition of the differentiae. But synonymous things were precisely those with both the name in common and the same definition. Hence all the things called from substances and differentiae are so called synonymously.
Anything that is predicated of a secondary substance or a difference is predicated of the primary substance synonymously. Remember that Aristotle has already shown that both differences and secondary substances are predicated of primary substances. Primary substances are not predicated of anything. Furthermore, the definition of the secondary substances and the differences will also be predicated of the primary substance. Aristotle has previously shown that when a word is used with the same name and definition in two places, that word is used synonymously. Since the definition and name of the difference and the secondary substance can be predicated of the primary substance, the meaning of it is the same in both places.
The significance of this claim is that when we call a human ‘animal’ and we call an ox ‘animal’, the word must mean the same thing in both cases. Similarly, when we say that these animals are different from other animals because they are mammals, mammals must mean the same thing to both things as well. This also means that anything predicated of secondary substances or differences is predicated of primary substances using the same meaning. For example, “men live” and “Tim, who is a man, lives” use the same meaning for ‘live’ because Tim is a man.
Next, Aristotle shows that substance is a unity.