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cf., ST.I.Q29.A3.Rep2; ST.I.Q29.A3.Rep4; ST.III.Q2.A2; Sent.I.D25.Q1.A1; QDePot.Q9.A2; QDeUni.A1
Old Article I wrote can be read here: https://www.christianbwagner.com/post/person-in-trinitarian-theology
There are two ways in which a term can be defined according to Aristotle, via a descending method or via an ascending method. Even then, the ascending method is used to confirm the definition given via the descending method. Here, St. Thomas first gives the definition via the descending method, then, in the next article, uses the ascending method to confirm the method.
Briefly put, the descending method works from the "top down," first finding the "generic term" that is most proximate to the thing that needs defining (e.g., if we wish to define "man," we would need to first determine that the generic term, i.e., the genus, is "animal"). Then, we discover that which is the root predication which distinguishes him most properly from the other "half" of the genus (in our case, the predication of "rationality" distinguishes him from other animals). Thus, by combining genus + specific difference, we are able to come to a definition.
Yet, there is another way we can do it. For, it is often hard to determine what the most proximate genus of some thing is and even harder to find out what the specific difference is...yet, we can determine both of these from what is better known to us. Here, we can compare the term with a number of different examples of things that are like it. Further, we can look at what is common between them (in order to determine the genus) and what differentiates them (in order to determine the difference). In the case of the latter, we often get a number of different predications (for example, in man, we discover that man "talks," "laughs," "reasons," and is "social"...all of which differentiate him from other animals). This is because there is not only a "specific difference," but also certain "properties," which flow from the essence...thus, there is a huge difference between a man being blue (which is not at all connected with his essence, but is completely accidental) and a man being able to talk (which directly flows from his rationality). Thus, we often reach a number of differentiating predications, and the role of the philosopher is to find out which of these predications are merely properties and which is the specific difference from which these properties flow. It is this method of comparison and differentiation that is the "ascending method."
In the first article, St. Thomas, taking from Boethius, uses the descending method to provide us with a "real definition" of person, and then, in the second article, uses the ascending method in order to confirm the definition.
St. Thomas begins, as in all such analyses with the "nominal definition." For, it is from the nominal definition that we come to arrive at the real definition. Here, he takes the nominal definition to be an "individual."