Aristotle says in his Categories,  “Substance, in the truest and primary and most definite sense of the word, is that which is neither predicable of a subject nor present in a subject.” Earlier in the text, he explains these characteristics:

Of things themselves some are predicable of a subject, and are never present in a subject. Thus ‘man’ is predicable of the individual man, and is never present in a subject.

By being ‘present in a subject’ I do not mean present as parts are present in a whole, but being incapable of existence apart from the said subject.

Some things, again, are present in a subject, but are never predicable of a subject. For instance, a certain point of grammatical knowledge is present in the mind, but is not predicable of any subject; or again, a certain whiteness may be present in the body (for colour requires a material basis), yet it is never predicable of anything.

Other things, again, are both predicable of a subject and present in a subject. Thus while knowledge is present in the human mind, it is predicable of grammar.

There is, lastly, a class of things which are neither present in a subject nor predicable of a subject, such as the individual man or the individual horse. But, to speak more generally, that which is individual and has the character of a unit is never predicable of a subject. Yet in some cases there is nothing to prevent such being present in a subject. Thus a certain point of grammatical knowledge is present in a subject.

It is not clear from this how Aristotle would deal with sentences like, “That man is Socrates,” where something individual seems to be predicated of a subject. Nonetheless, we can at least see what he means: things which are “predicable of a subject” refer to common terms and ideas, while individual realities are not said to be predicable of a subject. And something is “present in a subject” when it is a quality, property, or attribute of a thing. So by substance he refers to individual realities which are not qualities or attributes of other things.

Why does he say, “By being ‘present in a subject’ I do not mean present as parts are present in a whole, but being incapable of existence apart from the said subject”? Later in the text he explains this:

The fact that the parts of substances appear to be present in the whole, as in a subject, should not make us apprehensive lest we should have to admit that such parts are not substances: for in explaining the phrase ‘being present in a subject’, we stated’ that we meant ‘otherwise than as parts in a whole’.

In other words, he does not want to say that my arms and legs are “not substances.” Not because they are individual things in the way that I myself am an individual thing, but because neither do they seem to be merely qualities that I happen to have. The word “substance” itself is taken from the idea of “standing under,” and this is related to his general definition. Substances are the things that have everything else predicated of them or present in them, and in this sense they “stand under” those other things. And my arm seems to be a part of what stands under on an approximately equal level: my arm can be colored just as I can be colored, and so on.

Someone could object: if Socrates is a white man, then isn’t “white” a part of the whole, namely the “white man”? If this is the case, the Aristotle has no basis for his distinction between substantial parts and other things which are present in a subject.

But it is not difficult to respond to this. I tried to give a formal explanation of the idea of whole and part earlier. And as I said there, the whole must be distinct from each of the parts. I am not my legs, my arms, my torso, or my head. But in the case of Socrates, he is both the man and the white man. So “man” cannot be a part of the whole “white man”, since the man actually is the white man.

Nonetheless, it is possible to reformulate the objection to make it more problematic. Instead of suggesting that “white” and “man” are two parts of a whole, we could say that “whiteness” and “individual humanity” (or possibly “individuality” and “humanity”) are parts of Socrates. Socrates is not any of these things as such, and consequently we cannot refute the idea that they are parts as we did in the previous case.

I doubt that there can be any formal refutation of this idea, because if we could, we could refute the definition, “man is a rational animal,” in a similar way. In fact, if we add nothing to the idea of part and whole as we have defined them, it is not even false that “whiteness” and “humanity” can be parts of a whole. They can in fact be parts of a whole. It is just not the kind of whole that we are interested in when we ask the question, “what is that thing?” And this is the question that Aristotle is interested in when he speaks of substance, and consequently that kind of whole and part is not the kind he spoke about when he said that being “present in” a subject means “otherwise than as a part.”