Aristotle has just finished discussing numerical identity and identity in general. Now he is going to discuss two proofs for the division of dialectic questions by the four categories mentioned earlier. He does this in Topics, Book 1.
Of sameness then, as has been said, three types are to be distinguished. Now one way to confirm that the elements mentioned above are those out of which and through which and to which arguments proceed, is by induction- for if any one were to survey propositions and problems one by one, it would be seen that each was formed either from the definition of something or from its property or from its genus or from its accident. Another way to confirm it is through deduction. For every predicate of a subject must of necessity be either convertible with its subject or not: and if it is convertible, it would be its definition or property, for if it signifies the essence, it is the definition- if not, it is a property for this was what a property is, viz. what is predicated convertibly, but does not signify the essence. If, on the other hand, it is not predicated convertibly of the thing, it either is or is not one of the terms contained in the definition of the subject- and if it is one of those terms, then it will be the genus or the differentia, inasmuch as the definition consists of genus and differentiae- whereas, if it is not one of those terms, clearly it would be an accident, for accident was said to be what belongs to a subject without being either its definition or its genus or a property.
We have previously proven that there are three types of identity. Now there are two ways to confirm the previous division of dialectic questions into four categories. The first way is by induction. We could simply examine any dialectic question and show that the question is about a definition, a property, a genus or an accident. The second way is by deduction. Everything true of something must be true of that thing and only that thing necessarily convertible or not. This is necessarily true. If something is convertible, then it is either what that thing is an essence or definition or it is not and is therefore a property. If something is not convertible, then it is either one of the terms of the definition or it is not. The terms of the definition are the genus (what kind of thing is it) and the differentia (what makes it different from all other things belonging to that genus). Since the differentia is always a property of the thing, if something is not convertible, then it is the genus. The only remaining possibility is that something is not convertible and not in the definition. In that case it is an accident. Therefore, all dialectical questions are either of definitions, properties, genus or accidents.
In this passage Aristotle describes predicates and subjects. I have avoided using those words until now, when an explanation is absolutely necessary. A subject is what is being described. A predicate is how it is described. So if we say that a person is over there, a person is the subject, and over there is the predicate. Talking about predicates is another way of speaking that means the same thing as talking about dialectical questions. We are not just speaking about language but we are speaking about the way things in the world are.
Next, Aristotle will discuss how this division applies to the categories of predication.