The Four Pre-Categories from Ch. I an II of Aristotle's "Categories"
Stephen Alexander Beach
"Ὁμώνυμα λέγεται ὧν ὄνομα μόνον κοινόν, ὁ δὲ κατὰ τοὔνομα λόγος τῆς οὐσίας ἕτερος,"
"They are said of themselves homonomous, being the common name only. But according to the name the statement of essence differing."
Aristotle's work on the method of logic functions something like the process by which the mind begins to put limits on things in order to proceed to understanding them in a definition. It is the art and method, then, by which we come to know the real as a human knower, and can be summed up in the continuous question of Plato and Aristotle, What is it?. In the broadest sense we know undifferentiated "things", and as we continue to put limits we begin to assign "names" to the newly separated things, finally leading to a grasping of the inherent and real limits of things in themselves, called essences. So logic is studied by emphasizing the starting point of this journey of understanding as in the human person, while the starting point of metaphysics is in the real itself. This means that logic will take us from what is least known to us at first, to what is most known in itself last, while metaphysics will take us from what is most known in itself first to what is least known in itself last. Logic begins with the first principles of human knowing, not the first principles of being, as such. Therefore logic must begin in human language, and then proceed from language to the real. 
Chapter 1 - Equivocal, Univocal, and Derivative Speech
https://plato.stanford.edu/entries/aristotle-categories/
This can be reflected in the language that Aristotle chooses in his native Greek. He uses the word "onoma" "ὄνομα" to refer to a word which does not have a definite meaning yet. For example, "spring". You don't know what meaning I am referring to yet. I could mean water coming from the ground, a piece of metal, or a time of the year. When the word does have a associated definition that it can properly be called a word, or "logos" "λογος" in Greek. Likewise, when this definition is given, Aristotle calls it a "statement of essence" or "ὁ λογος τῆς οὐσίας". To illustrate this transition from the merely human and grammatical to the real and metaphysical we could also make a comparison between what Aristotle calls the "paronymous" or "derivative" and analogy. They are essentially the same, but with paronymous referring first to language usage, while analogy referring more to the being of things.
When it comes to the word subject, Aristotle uses the word "hupokeimena" or literally "that which lies under". When Aristotle talks of predication, he uses the word "lego" or "to speak" of something.
Chapter 1 - Equivocal, Univocal, and Derivative Speech
Aristotle begins by making a distinction between equivocal speech, univocal speech, and derivative speech. Equivocal speech is when things can have a common name and yet have different definitions. Univocal speech is when the name of things and their definitions are the same. He gives the example of a man and an ox. Both can be called animal, and when their definitions are explained of why they are animals they would be the same between both. Derivative speech refers to things which take an aspect of name from another thing, though are not that thing. He gives the example of a courageous man taking the name from the virtue of courage, yet they are not the same with one another.
Ομώνυμα λέγεται ὧν ὄνομα μόνον κοινόν, ὁ δὲ κατὰ τοὔνομα λόγος τῆς οὐσίας ἕτερος, οἷον ζῷον ὅ τε ἄνθρωπος καὶ τὸ γεγραμμένον· τούτων γὰρ ὄνομα μόνον κοινόν, ὁ δὲ κατὰ τοὔνομα λόγος τῆς οὐσίας ἕτερος· ἐὰν γὰρ ἀποδιδῷ τις τί ἐστιν αὐτῶν ἑκατέρῳ τὸ ζῴῳ εἶναι, ἴδιον ἑκατέρου λόγον ἀποδώσει. συνώνυμα δὲ λέγεται ὧν τό τε ὄνομα κοινὸν καὶ ὁ κατὰ τοὔνομα λόγος τῆς οὐσίας ὁ αὐτός, οἷον ζῷον ὅ τε ἄνθρωπος καὶ ὁ βοῦς· τούτων γὰρ ἑκάτερον κοινῷ ὀνόματι προσαγορεύεται ζῷον, καὶ ὁ λόγος δὲ τῆς οὐσίας ὁ αὐτός· ἐὰν γὰρ ἀποδιδῷ τις τὸν ἑκατέρου το λόγον τί ἐστιν αὐτῶν ἑκατέρῳ τὸ ζῴῳ εἶναι, τὸν αὐτὸν λόγον ἀποδώσει. παρώνυμα δὲ λέγεται ὅσα ἀπό τινος διαφέροντα τῇ πτώσει τὴν κατὰ τοὔνομα προσηγορίαν ἔχει, οἷον ἀπὸ τῆς γραμματικῆς ὁ γραμματικὸς καὶ ἀπὸ τῆς ἀνδρείας ὁ ἀνδρεῖος.
Chapter 2 - Subjects and Predicates
Here he makes a distinction between simple and composite "forms of speech". The simple would be solely a concept, like "man". Composite would be the union of the thing with predicate, "the man is running." When speaking of things as subjects (hupokeimenon - that which lies under, such as a universal truth lying under many appearances, or a unified subject) and predicates (lego - to say), one can speak of them in four ways. (1) predicable of a subject, but never present in a subject (2) present in a subject but never predicable of a subject (3) both predicable of a subject and present in a subject (4) Neither present in a subject nor predicable of a subject.
Τῶν λεγομένων τὰ μὲν κατὰ συμπλοκὴν λέγεται, τὰ δὲ ἄνευ συμπλοκῆς. τὰ μὲν οὖν κατὰ συμπλοκήν, οἷον ἄνθρωπος τρέχει, ἄνθρωπος νικᾷ· τὰ δὲ ἄνευ συμπλοκῆς, οἷον ἄνθρωπος, βοῦς, τρέχει, νικά.
Τῶν ὄντων τὰ μὲν καθ ̓ ὑποκειμένου τινὸς λέγεται, ἐν ὑποκειμένῳ δὲ οὐδενί ἐστιν, οἷον ἄνθρωπος καθ ̓ ὑποκειμένου μὲν λέγεται τοῦ τινὸς ἀνθρώπου, ἐν ὑποκειμένῳ δὲ οὐδενί ἐστιν· τὰ δὲ ἐν ὑποκειμένῳ μέν ἐστι, καθ ̓ ὑποκειμένου δὲ οὐδενὸς λέγεται, ἐν ὑποκειμένῳ δὲ λέγω ὃ ἔν τινι μὴ ὡς μέρος ὑπάρχον ἀδύνατον χωρὶς εἶναι τοῦ ἐν ᾧ ἐστίν, οἷον ἡ τὶς γραμματικὴ ἐν ὑποκειμένῳ μέν ἐστι τῇ ψυχῇ, καθ ̓ ὑποκειμένου δὲ οὐδενὸς λέγεται, καὶ τὸ τὶ λευκὸν ἐν ὑποκειμένῳ μέν ἐστι τῷ σώματι, ἅπαν γὰρ χρῶμα ἐν σώματι, καθ ̓ ὑποκειμένου δὲ οὐδενὸς λέγεται· τὰ δὲ καθ ̓ ὑποκειμένου τε λέγεται καὶ ἐν ὑποκειμένῳ ἐστίν, οἷον ἡ ἐπιστήμη ἐν ὑποκειμένῳ μέν ἐστι τῇ ψυχῇ, καθ ̓ ὑποκειμένου δὲ λέγεται τῆς γραμματικῆς· τὰ δὲ οὔτε ἐν ὑποκειμένῳ ἐστὶν οὔτε καθ ̓ ὑποκειμένου λέγεται, οἷον ὁ τὶς ἄνθρωπος ἢ ὁ τὶς ἵππος, οὐδὲν γὰρ τῶν τοιούτων οὔτε ἐν ὑποκειμένῳ ἐστὶν οὔτε καθ ̓ ὑποκειμένου λέγεται· ἁπλῶς δὲ τὰ ἄτομα καὶ ἓν ἀριθμῷ κατ ̓ οὐδενὸς ὑποκειμένου λέγεται, ἐν ὑποκειμένῳ δὲ ἔνια οὐδὲν κωλύει εἶναι· ἡ γὰρ τὶς γραμματικὴ τῶν ἐν ὑποκειμένῳ ἐστίν.
(1) predicable of a subject, but never present in a subject - Here Aristotle refers to the universal essences of things, such as "man," for example. They can be predicated of each individual man, but yet no individual man will contain all that is implied in the essence of man itself. Thus we can never say that "man-ness" as such exists in any man at all.
(2) present in a subject but never predicable of a subject - This refers to an accident which exists in an individual subject, such as "thinness" in this particular man, Joey. But we cannot predicate thinness to all men as being essential to being man. It will never exist on its own, but only withing an already existing thing.
(3) both predicable of a subject and present in a subject - Here we have a proper accident, or a universal accident. It can be predicated of a universal, such as all men having color, for example. Likewise, it also exists in the individual man, such as Joey being white.
(4) Neither present in a subject nor predicable of a subject - There are singular substances, such as Socrates, for example, which we can not predicate of all men, nor can we say that Socrates exists in Socrates as in part, like an accident, but Socrates is a whole, a substance, and therefore we don't say that Socrates is in Socrates, but that Socrates is Socrates.
The chart below should make this clearer.
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https://plato.stanford.edu/entries/aristotle-categories/
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