Aristotle has described the various ways that two things could be identical. Now he goes on to describe the various ways in which something is numerically identical. He does this in Topics, Book 1.
It seems that things numerically one are called the same by everyone with the greatest degree of agreement. But this too is apt to be rendered in more than one sense; its most literal and primary use is found whenever the sameness is rendered by a name or definition, as when a cloak is said to be the same as a doublet, or a two-footed terrestrial animal is said to be the same as a man; a second sense is when it is rendered by a property, as when what can acquire knowledge is called the same as a man, and what naturally travels upward the same as fire; while a third use is found when it is rendered in reference to some accident, as when the creature who is sitting, or who is musical, is called the same as Socrates. For all these are meant to signify numerical unity. That what I have just said is true may be best seen where one form of appellation is substituted for another. For often when we give the order to call one of the people who are sitting down, indicating him by name, we change our description, whenever the person to whom we give the order happens not to understand us; he will, we think, understand better from some accidental feature; so we bid him call to us the man who is sitting or who is conversing—clearly supposing ourselves to be indicating the same object by its name and by its accident.
It seems that numerical identity is the closest kind of identity there is. However, things are numerically identical in three ways. First, we say that this thing is the same thing that the definition of that thing says it is. Second, we say that this thing is the same thing as the thing that has a particular property. For example, humans can learn abstractly, so we can speak of human beings as abstract learners. Thirdly, we say that this thing is the same thing as something that has an accident. So we can speak of a particular person as “that person on my right”. All of these things are different ways of speaking of numerical identity. That this is true is obvious from an ordinary feature of life. Sometimes when a teacher is recording attendance in class, one of the students will not answer to their name. So we change our description to some accidental feature of them – you next to James, or the person who is talking right now. This shows that we can indicate the same thing by using the name and a description.
Next, Aristotle describes two ways of proving that the previous division of questions into four categories leaves nothing out.