ficino wrote:If there is a genus F, then is it the case that for all x, if x is F, x must be in the genus F?
Genus, differentia, species and the likes are found in Aristotelian logic. Aristotelian logic has now been superseded by First Order Logic. So, the following discussion must be construed in an Aristotelian logical context. In modern logic, concepts such as predicate is different from that in Aristotelian logic. In a modern context, a predicate is construed as a function, where as in ancient logic it is a term. There are other major differences as well such as the use of quantifiers and truth functional connectives in modern logic, all of which are absent in ancient logic.
A genus is a larger class containing under it smaller classes. In the statement ‘Man is a mammal’ mammal is a genus with respect to man.
A species is a smaller class contained under a larger one. In the statement: ’Aristotle is a man’, man is species with respect to Aristotle.
When one talks of genus, one is also talking of species and individuals. An Individuals belongs in/under a species and a species belongs in/under a genus. So if one says that x belongs in/under the genus F, then either x is a species or x is an individual. If x is an individual, then it must also belong in a species, and the species will belong in a genus.
For an individual man such as for example Aristotle, it can be said that he belonged to the species man, and as man is a species which belong to the genus mammal, Aristotle also belonged in the genus mammal.
But as for God, the Almighty, such things cannot be said. For God, the Almighty is Absolute Oneness, and there is and can be no species to which He belongs. Since there is no-one like Him, the Almighty, therefore He does not belong in/under any species. As He does not belong to any species, there can also be no genus in which He would belong.
The genus and the differentia together make up the definition of a species. The definition of something expresses the essence of that something. Now according to Aristotle thoughts, only species can be defined. From Stanford Encyclopedia of Philosophy in the entry on Aristotle’s logic, we have this:
Since a definition defines an essence, only what has an essence can be defined. What has an essence, then? That is one of the central questions of Aristotle’s metaphysics; once again, we must leave the details to another article. In general, however, it is not individuals but rather species (eidos: the word is one of those Plato uses for “Form”) that have essences. A species is defined by giving its genus (genos) and its differentia (diaphora): the genus is the kind under which the species falls, and the differentia tells what characterizes the species within that genus. As an example, human might be defined as animal (the genus) having the capacity to reason (the differentia).
Reference link:
https://plato.stanford.edu/entries/aristotle-logic/#Def
Now from what Aristotle says in the categories, it is clear that the notion of a genus and a species applies to more than one thing:
A substance-that which is called a substance most strictly, primarily, and most of all--is that which is neither said of a subject nor in a subject, e.g. the individual man or the individual horse. The species in which the things primarily called substances are, are called secondary substances, as also are the genera of these species. For example, the individual man belongs in a species, man, and animal is a genus of the species; so these-both man and animal-are called secondary substances.
(…)
Every substance seems to signify a certain 'this'. As regards the primary substances, it is indisputably true that each of them signifies a certain 'this'; for the thing revealed is individual and numerically one. But as regards the secondary substances, though it appears from the form of the name-when one speaks of man or animal—that a secondary substance likewise signifies a certain 'this', this is not really true; rather, it signifies a certain qualification, for the subject is not, as the primary substance is, one, but man and animal are said of many things. However, it does not signify simply a certain qualification, as white does. White signifies nothing but a qualification, whereas the species and the genus mark off the qualification of substance-they signify substance of a certain qualification. (One draws a wider boundary with the genus than with the species, for in speaking of animal one takes in more than in speaking of man.) [Categories chapter 5]
So, the genera and species are only said of many things.
But since God, the Almighty, is Absolute Oneness, and there is none like Him, therefore there is and can be no species to which He belongs. But that now does not mean that nothing can be predicated of God, the Almighty. But what can be said is this: that which is predicated of God, the Almighty cannot be said to be a genus or a differentiae or species.
Genus, species and differentia belong to the class of things known as the predicables in ancient logic. The other predicables are: property and accident.
ficino wrote:…if we say there is a genus of movers, the First Unmoved Mover must be in the genus of movers if it is to be a mover.
If you say that there is a genus F, then you are saying that there are species which are defined with F as their genus. Of such species which belong in genus F, F is predicated of them. Every genus is predicated of the species which fall under it. This means that every genus is a predicate. But not every predicate is a genus. There are predicates which are accidents and properties as well. So you should not confuse these two notions, namely predicate and genus.
Anyway, that is why, I would, as well, not include God, the Almighty in any genus or species.