Aristotle has just divided the area of dialectic up into four parts. Now he begins a short discussion of sameness and difference. He will relate this discussion to the four parts of dialectic. He does this in Topics, Book 1.
First of all we must determine the number of ways we talk of sameness. Sameness would be generally regarded as falling, roughly speaking, into three divisions. We generally apply the term numerically or specifically or generically numerically in cases where there is more than one name but only one thing, e.g. doublet and cloak- specifically, where there is more than one thing, but they present no differences in respect of their species, as one man and another, or one horse and another- for things like this that fall under the same species are said to be specifically the same. Similarly, too, those things are called generically the same which fall under the same genus, such as a horse and a man. It might appear that the sense in which water from the same spring is called the same water is somehow different and unlike the senses mentioned above- but really such a case as this ought to be ranked in the same class with the things that in one way or another are called the same in view of unity of species. For all such things seem to be of one family and to resemble one another. For the reason why all water is said to be specifically the same as all other water is because of a certain likeness it bears to it, and the only difference in the case of water drawn from the same spring is this, that the likeness is more emphatic: that is why we do not distinguish it from the things that in one way or another are called the same in view of unity of species.
We will now explain the different ways that we talk about sameness. There are three of them in general: numerical, species and generic. Numerical identity happens when we speak of one thing but use more than one name for it. An example is a doublet and a cloak. Species identity happens when two things are the same kind of thing. For example, Aristotle and Plato are both human or their horses are both horses. Finally, generic identity occurs when two kinds of things belong to the same category. For example, humans and horses are both animals. Aristotle argues that our language does not show any evidence of an exception to this. Although we say that two glasses of water from the same spring are the same water, this is not a fourth category. This is just a way of saying that all water is the same kind of thing species identity. We are emphasizing this by pointing out that water from that spring is the same as other water from that spring. This sort of sameness is more obvious in this case.
While I do not disagree with Aristotles analysis of identity, I think that his response to the same spring argument is incorrect. We should point out that water sometimes tastes different but water from the same source tastes the same. The sameness of taste is because the accidents are the same. So what is really being discussed is similarity of accident which is a kind of species. Modern philosophers, however, do add other categories of identity.
Next, Aristotle discusses numerical identity.