Aristotle Metaphysics IV4 1007a21-1007b18

• First, know that in order for us to understand Aristotle's argument here, we must back up at least to 1006a1 and understand a bit of the context. Please go read that at your leisure. You needn't understand every bit, but it will provide some context.
  
• Aristotle is discussing people who, when asked "what is this?" (pointing to a man), answer "It is a man" but also "It is not a man." Why would they answer both of those things? Because they think that "it is a musical thing" and "it is a be-trousered thing" and "it is a hairy thing" etc. and NONE of those things are identical to "being a man." Therefore, "it is a man and it is not a man" is an answer that they think is right (Aristotle disagrees). Aristotle wants to show that such people cannot possibly be coherent or successfully refer to anything without contradicting themselves in a way that they themselves must agree is a contradiction (they won't admit right off that "it is a man and it is not a man" is a contradiction).  Along the way, in the following paragraph, Aristotle argues that things are not just associations of accidental properties: there must be something that an accident is an accident of, and that cannot be merely another accident. He is arguing that there must be essences (which amounts to the same thing as saying that there must be substances, he thinks).
  
• And in general those who use this argument do away with substance and essence. For they must say that all attributes are accident, and that there is no such thing as being essentially man or animal. For if there is to be any such thing as being essentially man, this will not be being not-man or not being man (yet these are negations of it); for there was some one thing which it meant, and this was the substance of something. And denoting the substance of a thing means that the essence of the thing is nothing else. But if its being essentially man is to be the same as either being essentially not-man or essentially not being man, then its essence will be something else. Therefore our opponents must say that there cannot be such a definition of anything, but that all attributes are accidental; for this is the distinction between substance and accident--white is accidental to man, because though he is white, whiteness is not his essence. But if all statements are accidental, there will be nothing primary about which they are made, if the accidental always implies predication about a subject. The predication, then , must go on ad infinitum. But this is impossible; for not even more than two terms can be combined. For an accident is not an accident of an accident, unless it be because both are accidents of the same subject. I mean, for instance, the white is musical and the latter is white, only because both are accidental to man. But Socrates is musical, not in this sense, that both terms are accidental to something else. Since then some predicates are accidental in this and some in that sense, those which are accidental in the latter sense, in which white is accidental to Socrates, cannot form an infinite series in the upward direction--e.g. Socrates the white has not yet another accident; for no unity can be got out of such a sum. Nor again will white have another term accidental to it, e.g. musical. For this is no more accidental to that than that is to this; and at the same time we have drawn the distinction, that while some predicates are accidental in this sense, others are so in the sense in which musical is accidental to Socrates; and the accident is an accident of an accident not in cases of the latter kind, but only in cases of the other kind, so that not all terms will be accidental. There must, then, even in this case be something that denotes substance. And it has been shown that, if this is so, contradictories cannot be predicated at the same time.